Step-by-Step Guidance for Beginning-Intermediate Algebra Questions

{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"{\"type\":\"doc\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q1. Find the slope and the y-intercept of the line: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -\\\\frac{6}{7}x - \\\\frac{2}{3}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Linear Equations in Slope-Intercept Form\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to identify the slope and y-intercept from a linear equation written in the form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Slope (m):\"},{\"type\":\"text\",\"text\":\" The coefficient of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" in \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"y-intercept (b):\"},{\"type\":\"text\",\"text\":\" The constant term in \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\"; the point where the line crosses the y-axis (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x=0\"}},{\"type\":\"text\",\"text\":\").\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the given equation to the standard form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify the coefficient of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" as the slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify the constant term as the y-intercept \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"b\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Express the y-intercept as the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, b)\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Slope: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-\\\\frac{6}{7}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"y-intercept: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, -\\\\frac{2}{3})\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The slope is the coefficient of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\", and the y-intercept is the constant term.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q2. Solve: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|x - 3| = 22\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Absolute Value Equations\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to solve equations involving absolute values, which often have two possible solutions.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Absolute Value:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|a| = b\"}},{\"type\":\"text\",\"text\":\" means \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a = b\"}},{\"type\":\"text\",\"text\":\" or \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a = -b\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up two equations: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x - 3 = 22\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x - 3 = -22\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve each equation for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" by adding $3"},{"type":"inlineMath","attrs":{"latex":" to both sides.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The solution set is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\{25, -19\\\\}\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Both values satisfy the original absolute value equation.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q3. Divide: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{\\\\left(x^4-7x^2+10\\\\right)}{\\\\left(x^2-2\\\\right)}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Polynomial Division\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to divide polynomials, either by long division or synthetic division.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Polynomial Division:\"},{\"type\":\"text\",\"text\":\" Dividing one polynomial by another, similar to long division with numbers.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the division: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^4 - 7x^2 + 10\"}},{\"type\":\"text\",\"text\":\" divided by \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2 - 2\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Divide the leading term \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^4\"}},{\"type\":\"text\",\"text\":\" by \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2\"}},{\"type\":\"text\",\"text\":\" to get the first term of the quotient.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2 - 2\"}},{\"type\":\"text\",\"text\":\" by this term and subtract from the original polynomial.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2 - 7 + \\\\dfrac{24}{x^2 - 2}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The quotient is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2 - 7\"}},{\"type\":\"text\",\"text\":\" with a remainder of $24"}},{"type":"text","text":".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q4. Rewrite with positive exponents: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{1}{x^{-3/4}}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Exponents and Radicals\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of negative exponents and how to rewrite expressions with only positive exponents.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Negative Exponent Rule:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^{-a} = \\\\dfrac{1}{x^a}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Reciprocal Rule:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{1}{x^{-a}} = x^a\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recognize that a negative exponent in the denominator becomes positive in the numerator.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Apply the reciprocal rule to rewrite the expression with a positive exponent.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^{3/4}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Negative exponents in the denominator become positive in the numerator.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q5. Find an equation of the line with slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m = 9\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" passing through the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(3, 1)\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Writing Equations of Lines\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to write the equation of a line given a slope and a point.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Point-Slope Form:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y - y_1 = m(x - x_1)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Slope-Intercept Form:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug the given point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(3, 1)\"}},{\"type\":\"text\",\"text\":\" and slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m = 9\"}},{\"type\":\"text\",\"text\":\" into the point-slope form: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y - 1 = 9(x - 3)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Expand the equation to get \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 9x - 27 + 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify to write the equation in slope-intercept form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 9x - 26\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The equation is in slope-intercept form, passing through \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(3, 1)\"}},{\"type\":\"text\",\"text\":\" with slope $9"},{"type":"inlineMath","attrs":{"latex":".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q6. Express in terms of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"i\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2 - \\\\sqrt{-28}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Complex Numbers\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to simplify square roots of negative numbers using the imaginary unit \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"i\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Imaginary Unit:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"i = \\\\sqrt{-1}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Simplifying:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{-a} = i\\\\sqrt{a}\"}},{\"type\":\"text\",\"text\":\" for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a > 0\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rewrite \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{-28}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"i\\\\sqrt{28}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{28}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{4 \\\\times 7} = 2\\\\sqrt{7}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Combine the terms to write the expression in the form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a + bi\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2 - 2i\\\\sqrt{7}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The expression is written in the form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a + bi\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q7. Solve: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{z - 8}{z - 9} = \\\\dfrac{1}{z - 9}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rational Equations\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to solve equations involving rational expressions.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Rational Equation:\"},{\"type\":\"text\",\"text\":\" An equation involving fractions whose numerators and/or denominators contain variables.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Since the denominators are the same, set the numerators equal: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z - 8 = 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z\"}},{\"type\":\"text\",\"text\":\" by adding $8"}},{"type":"text","text":" to both sides.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check that \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z\"}},{\"type\":\"text\",\"text\":\" does not make the denominator zero (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z \\\\neq 9\"}},{\"type\":\"text\",\"text\":\").\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 9\"}},{\"type\":\"text\",\"text\":\" is not valid because it makes the denominator zero. The solution is "},{"type":"inlineMath","attrs":{"latex":"z = 9"}},{"type":"text","text":" is extraneous, so there is no solution.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q8. Factor: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"147v^2 - 192\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Factoring Polynomials\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to factor a difference of squares or factor out the greatest common factor (GCF).\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Difference of Squares:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a^2 - b^2 = (a - b)(a + b)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"GCF:\"},{\"type\":\"text\",\"text\":\" The largest factor common to all terms.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Look for a GCF in \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"147v^2\"}},{\"type\":\"text\",\"text\":\" and $192"},{"type":"inlineMath","attrs":{"latex":".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor out the GCF.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if the remaining expression is a difference of squares or can be factored further.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"147v^2 - 192 = 49v^2 - 64 \\\\times 3 = 3(49v^2 - 64)\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Then factor \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"49v^2 - 64\"}},{\"type\":\"text\",\"text\":\" as a difference of squares: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(7v - 8)(7v + 8)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q9. Find the equation of the line through \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-4, -5)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-12, -15)\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Equation of a Line from Two Points\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to find the equation of a line given two points.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Slope Formula:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m = \\\\dfrac{y_2 - y_1}{x_2 - x_1}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Point-Slope Form:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y - y_1 = m(x - x_1)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Label the points: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x_1, y_1) = (-4, -5)\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x_2, y_2) = (-12, -15)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m\"}},{\"type\":\"text\",\"text\":\" using the formula above.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the point-slope form with one of the points and the calculated slope.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify to slope-intercept form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = \\\\frac{5}{4}x\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The line passes through both points and has slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{5}{4}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q10. Write the equation of the line through \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(5, 8)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and perpendicular to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"6x + y = 9\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Perpendicular Lines\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to find the equation of a line perpendicular to a given line and passing through a specific point.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Slope of Perpendicular Lines:\"},{\"type\":\"text\",\"text\":\" If a line has slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m\"}},{\"type\":\"text\",\"text\":\", a perpendicular line has slope \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-\\\\frac{1}{m}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Point-Slope Form:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y - y_1 = m(x - x_1)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rewrite \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"6x + y = 9\"}},{\"type\":\"text\",\"text\":\" in slope-intercept form to find its slope.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the negative reciprocal of this slope for the perpendicular line.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the point-slope form with \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(5, 8)\"}},{\"type\":\"text\",\"text\":\" and the perpendicular slope.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = mx + b\"}},{\"type\":\"text\",\"text\":\" form.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -6x + 38\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The perpendicular slope is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-6\"}},{\"type\":\"text\",\"text\":\", and the line passes through \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(5, 8)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q11. Rationalize the denominator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{4}{4 - \\\\sqrt{3}}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rationalizing Denominators\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to eliminate radicals from the denominator of a fraction.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Conjugate:\"},{\"type\":\"text\",\"text\":\" For \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a - b\"}},{\"type\":\"text\",\"text\":\", the conjugate is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a + b\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Rationalizing:\"},{\"type\":\"text\",\"text\":\" Multiply numerator and denominator by the conjugate of the denominator.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply numerator and denominator by \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"4 + \\\\sqrt{3}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Expand the numerator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"4(4 + \\\\sqrt{3})\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Expand the denominator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(4 - \\\\sqrt{3})(4 + \\\\sqrt{3})\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify the denominator using the difference of squares formula.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{16 + 4\\\\sqrt{3}}{13}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The denominator is rationalized by multiplying by the conjugate.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q12. For \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 2x^2 + 2x + 4\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", find (a) the vertex, (b) the line of symmetry, (c) the maximum or minimum value, and (d) graph the function.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Quadratic Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of the properties of quadratic functions, including finding the vertex, axis of symmetry, and minimum/maximum values.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Vertex Formula:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -\\\\frac{b}{2a}\"}},{\"type\":\"text\",\"text\":\" for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = ax^2 + bx + c\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Axis of Symmetry:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -\\\\frac{b}{2a}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Minimum/Maximum Value:\"},{\"type\":\"text\",\"text\":\" For \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a > 0\"}},{\"type\":\"text\",\"text\":\", the vertex is a minimum; for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a < 0\"}},{\"type\":\"text\",\"text\":\", it's a maximum.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a = 2\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"b = 2\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"c = 4\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-coordinate of the vertex: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -\\\\frac{2}{2 \\\\times 2}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug this \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" value into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" to find the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-coordinate of the vertex.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The axis of symmetry is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \"}},{\"type\":\"text\",\"text\":\" (the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-coordinate of the vertex).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Since \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a > 0\"}},{\"type\":\"text\",\"text\":\", the vertex is a minimum point.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Vertex: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\left(-\\\\frac{1}{2}, \\\\frac{7}{2}\\\\right)\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Line of symmetry: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -\\\\frac{1}{2}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) Minimum value: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{7}{2}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The graph is a parabola opening upwards.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q13. The length of a rectangle is twice the width. The area is 72\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" yd\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"^2. Find the length and width.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Area of Rectangles and Algebraic Equations\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to set up and solve equations based on geometric relationships.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Area of Rectangle:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"A = l \\\\times w\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Relationship:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"l = 2w\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Let \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"w\"}},{\"type\":\"text\",\"text\":\" be the width. Then length \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"l = 2w\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"l \\\\times w = 72\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"l = 2w\"}},{\"type\":\"text\",\"text\":\" into the equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2w \\\\times w = 72\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"w\"}},{\"type\":\"text\",\"text\":\" by dividing both sides by $2"}},{"type":"text","text":" and taking the square root.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"l\"}},{\"type\":\"text\",\"text\":\" by doubling \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"w\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Width: "},{"type":"inlineMath","attrs":{"latex":"6\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" yd, Length: $12\"}},{\"type\":\"text\",\"text\":\" yd\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"12 \\\\times 6 = 72\"}},{\"type\":\"text\",\"text\":\" yd\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"^2\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q14. A car travels 660\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"milesatacertainspeed.Ifthespeedhadbeen5\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" mph faster, the trip would have taken $1"}},{"type":"text","text":" hour less. Find the speed.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Distance, Rate, and Time Problems\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to set up and solve equations involving distance, rate, and time.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Distance Formula:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"d = rt\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Time:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\frac{d}{r}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Let \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r\"}},{\"type\":\"text\",\"text\":\" be the original speed. Then time taken is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\frac{660}{r}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"If the speed was \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r + 5\"}},{\"type\":\"text\",\"text\":\", the time would be \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{660}{r + 5}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{660}{r} - \\\\frac{660}{r + 5} = 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply both sides by \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r(r + 5)\"}},{\"type\":\"text\",\"text\":\" to clear denominators and solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Speed: $60"},{"type":"inlineMath","attrs":{"latex":" mph\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{660}{60} = 11\"}},{\"type\":\"text\",\"text\":\" hours, \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{660}{65} \\\\approx 10.15\"}},{\"type\":\"text\",\"text\":\" hours, difference is about $1"}},{"type":"text","text":" hour.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q15. Divide and simplify: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{y^2 - 1}{y} \\\\div \\\\dfrac{y - 1}{y + 1}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Operations with Rational Expressions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to divide and simplify rational expressions.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Division of Fractions:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{a}{b} \\\\div \\\\dfrac{c}{d} = \\\\dfrac{a}{b} \\\\times \\\\dfrac{d}{c}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Factoring:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y^2 - 1 = (y - 1)(y + 1)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rewrite the division as multiplication by the reciprocal: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{y^2 - 1}{y} \\\\times \\\\dfrac{y + 1}{y - 1}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y^2 - 1\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(y - 1)(y + 1)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify by canceling common factors in numerator and denominator.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{(y + 1)^2}{y}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"All possible factors have been canceled and the expression is simplified.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q16. Find the x-intercepts and y-intercepts of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = -x^2 + 4x + 12\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Intercepts of Quadratic Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to find where a quadratic function crosses the x-axis and y-axis.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"x-intercepts:\"},{\"type\":\"text\",\"text\":\" Set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 0\"}},{\"type\":\"text\",\"text\":\" and solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"y-intercept:\"},{\"type\":\"text\",\"text\":\" Set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\" and solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(0)\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-x^2 + 4x + 12 = 0\"}},{\"type\":\"text\",\"text\":\" and solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" (use quadratic formula if needed).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For the y-intercept, substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\" into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"x-intercepts: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-2, 0)\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(6, 0)\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"y-intercept: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, 12)\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q17. Graph the system of inequalities: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y \\\\geq x\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y \\\\leq -x + 14\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\". Find the coordinates of any vertices formed.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Systems of Linear Inequalities\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to graph inequalities and find the intersection (vertex) of the boundary lines.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Intersection Point:\"},{\"type\":\"text\",\"text\":\" Solve \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = x\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -x + 14\"}},{\"type\":\"text\",\"text\":\" simultaneously.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = x\"}},{\"type\":\"text\",\"text\":\" equal to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -x + 14\"}},{\"type\":\"text\",\"text\":\" to find the intersection point.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" and then for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Vertex: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(7, 7)\"}}]},{\"type\":\"par