College Algebra Exam Review: Step-by-Step Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Solve the inequality and write your answer in interval notation: -6x + 9 \\geq 3x - 18"}]},{"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 Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve linear inequalities and express the solution in interval notation."}]},{"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","text":"Linear inequality: An inequality involving a linear expression."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interval notation: A way to represent solution sets using intervals."}]}]}]},{"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":"Start by moving all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other. Subtract "},{"type":"inlineMath","attrs":{"latex":"3x"}},{"type":"text","text":" from both sides."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms to simplify the inequality."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by moving the constant term to the other side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by the coefficient of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":". Remember, if you divide by a negative number, the inequality sign reverses."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x \\leq 3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"In interval notation: "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, 3]"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We solved for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and remembered to flip the inequality when dividing by a negative."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Solve the inequality: 4(x - 5) < 2x + 6"}]},{"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 Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve and simplify linear inequalities."}]},{"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","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear inequality: An inequality with a variable to the first power."}]}]}]},{"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":"Apply the distributive property to expand "},{"type":"inlineMath","attrs":{"latex":"4(x - 5)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms to simplify."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by dividing both sides by the coefficient of $x$."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x < 13"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"In interval notation: "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, 13)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying and isolating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", you find the solution set."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Solve and graph the solution set: \\frac{x - 1}{4} \\geq 2"}]},{"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 Inequalities with Fractions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving fractions and to represent the solution graphically."}]},{"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","text":"To clear fractions, multiply both sides by the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Graphing solution sets: Use a number line to represent the solution."}]}]}]},{"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 both sides of the inequality by 4 to eliminate the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the resulting inequality."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by adding or subtracting as needed."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution in interval notation and consider how you would graph it on a number line."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x \\geq 9"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"In interval notation: "},{"type":"inlineMath","attrs":{"latex":"[9, \\infty)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides by 4, then solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Solve the inequality: -3(2x - 1) \\leq 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: Linear Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving distribution and negative coefficients."}]},{"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","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remember to reverse the inequality if dividing by a negative."}]}]}]},{"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":"Apply the distributive property to expand "},{"type":"inlineMath","attrs":{"latex":"-3(2x - 1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the inequality by combining like terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by moving constants to one side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by the coefficient of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", remembering to reverse the inequality if dividing by a negative."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x \\geq -2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"In interval notation: "},{"type":"inlineMath","attrs":{"latex":"[-2, \\infty)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Be careful with the sign when dividing by a negative number."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Solve the absolute value equation: |x + 8| = 5"}]},{"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 understanding of how to solve equations involving absolute value."}]},{"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","text":"Absolute value: "},{"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":"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 separate equations: "},{"type":"inlineMath","attrs":{"latex":"x + 8 = 5"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x + 8 = -5"}},{"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":"."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x = -13"}},{"type":"text","marks":[{"type":"bold"}],"text":" or "},{"type":"inlineMath","attrs":{"latex":"x = -3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are two solutions because the absolute value can be positive or negative."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Solve the absolute value inequality: |3x - 2| < 10"}]},{"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 Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving absolute value 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","text":"For "},{"type":"inlineMath","attrs":{"latex":"|A| < B"}},{"type":"text","text":", rewrite as "},{"type":"inlineMath","attrs":{"latex":"-B < A < 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":"Rewrite the inequality as "},{"type":"inlineMath","attrs":{"latex":"-10 < 3x - 2 < 10"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add 2 to all three parts of the inequality."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide all parts by 3 to solve for "},{"type":"inlineMath","attrs":{"latex":"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":"inlineMath","attrs":{"latex":"-\\frac{8}{3} < x < 4"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution as an open interval: "},{"type":"inlineMath","attrs":{"latex":"\\left(-\\frac{8}{3}, 4\\right)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Solve the absolute value inequality: -2|x - 4| \\geq -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: Absolute Value Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to manipulate and solve inequalities involving absolute value and negative coefficients."}]},{"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","text":"When dividing or multiplying both sides of an inequality by a negative, reverse the inequality sign."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"|A| \\leq B"}},{"type":"text","text":", rewrite as "},{"type":"inlineMath","attrs":{"latex":"-B \\leq A \\leq 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":"Divide both sides by -2, remembering to reverse the inequality sign."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the resulting absolute value inequality in its compound form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the compound inequality for "},{"type":"inlineMath","attrs":{"latex":"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":"inlineMath","attrs":{"latex":"-1 \\leq x \\leq 9"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution as a closed interval: "},{"type":"inlineMath","attrs":{"latex":"[-1, 9]"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Solve for x: 7(x - 2) - 5 = 3x + 11"}]},{"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"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve linear equations with variables on both sides."}]},{"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","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms and isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"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":"Expand "},{"type":"inlineMath","attrs":{"latex":"7(x - 2)"}},{"type":"text","text":" using the distributive property."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms on both sides of the equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by dividing both sides by the coefficient of $x$."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x = 4"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying and isolating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", you find the solution."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Solve for x: \\frac{2x}{3} + 4 = \\frac{x}{6} + 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: Linear Equations with Fractions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve linear equations involving fractions."}]},{"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","text":"To clear fractions, multiply both sides by the least common denominator (LCD)."}]}]}]},{"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":"Find the LCD for the denominators (3 and 6)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides of the equation by the LCD to eliminate fractions."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and solve for "},{"type":"inlineMath","attrs":{"latex":"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":"inlineMath","attrs":{"latex":"x = 6"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying both sides by 6 clears the denominators and allows you to solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Solve for x: 10 - 3(2x + 1) = x - 8"}]},{"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"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve linear equations with distribution and variables on both sides."}]},{"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","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms and isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"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":"Expand "},{"type":"inlineMath","attrs":{"latex":"-3(2x + 1)"}},{"type":"text","text":" using the distributive property."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms on both sides."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by dividing both sides by the coefficient of $x$."}]}]}]},{"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":"inlineMath","attrs":{"latex":"x = 3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying and isolating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", you find the solution."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Find the slope and y-intercept of the line: 5x - y = 10"}]},{"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 Standard Form"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to rewrite a linear equation in slope-intercept form and identify the slope and y-intercept."}]},{"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","text":"Slope-intercept form: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope ("},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"): The coefficient of "},{"type":"inlineMath","attrs":{"latex":"x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Y-intercept ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"): The constant term"}]}]}]},{"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 equation in the form "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}},{"type":"text","text":" by solving for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the slope ("},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":") and y-intercept ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":") from the equation."}]}]}]},{"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":"inlineMath","attrs":{"latex":"m = 5"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"b = -10"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After rearranging, the equation is "},{"type":"inlineMath","attrs":{"latex":"y = 5x - 10"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Find the slope and y-intercept of the line: y = -2x - 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: 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 an equation already in slope-intercept form."}]},{"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","text":"Slope-intercept form: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope ("},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"): The coefficient of "},{"type":"inlineMath","attrs":{"latex":"x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Y-intercept ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"): The constant term"}]}]}]},{"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 the slope ("},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":") as the coefficient of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the y-intercept ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":") as the constant term."}]}]}]},{"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":"inlineMath","attrs":{"latex":"m = -2"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"b = -3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The equation is already in slope-intercept form."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. Find the slope and y-intercept (if possible) of the line: x = 6"}]},{"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: Vertical and Horizontal Lines"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the properties of vertical 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","text":"Vertical line: "},{"type":"inlineMath","attrs":{"latex":"x = a"}},{"type":"text","text":" has an undefined slope and no y-intercept (unless it crosses the y-axis)."}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"x = 6"}},{"type":"text","text":" is a vertical line."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that vertical lines have undefined slope and do not have a y-intercept."}]}]}]},{"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: Slope is undefined; no y-intercept"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical lines do not cross the y-axis, so no y-intercept exists."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. Write the equation of the line through (1, -2) and (5, 6)"}]},{"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 Given 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","text":"Slope formula: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope-intercept form: "},{"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":"Calculate the slope "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" using the two points."}]}]},{"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":"Rearrange the equation into slope-intercept 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":"inlineMath","attrs":{"latex":"y = 2x - 4"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After finding the slope and using point-slope form, you get the equation in slope-intercept form."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. Write the equation of the line with slope "},{"type":"inlineMath","attrs":{"latex":"m = -4"}},{"type":"text","marks":[{"type":"bold"}],"text":" through (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: Equation of a Line Given Slope and Point"}]},{"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","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope-intercept form: "},{"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 slope and point into the point-slope form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and rearrange into slope-intercept 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":"inlineMath","attrs":{"latex":"y = -4x + 11"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute the values and solve for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Write an equation of the line parallel to "},{"type":"inlineMath","attrs":{"latex":"y = 3x - 7"}},{"type":"text","marks":[{"type":"bold"}],"text":" through "},{"type":"inlineMath","attrs":{"latex":"(-1, 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: Parallel Lines"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write the equation of a line parallel to a given line 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","text":"Parallel lines have the same slope."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"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":"Identify the slope of the given line ("},{"type":"inlineMath","attrs":{"latex":"m = 3"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the point-slope form with the given point and slope."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify to slope-intercept 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":"inlineMath","attrs":{"latex":"y = 3x + 7"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Parallel lines have the same slope as the original line."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Write an equation of the line perpendicular to "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{1}{2}x + 5"}},{"type":"text","marks":[{"type":"bold"}],"text":" through "},{"type":"inlineMath","attrs":{"latex":"(4, -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: Perpendicular Lines"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write the equation of a line perpendicular to a given line 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","text":"Perpendicular slopes are negative reciprocals: "},{"type":"inlineMath","attrs":{"latex":"m_1 \\cdot m_2 = -1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"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":"Find the negative reciprocal of the given slope ("},{"type":"inlineMath","attrs":{"latex":"m = \\frac{1}{2}"}},{"type":"text","text":" becomes "},{"type":"inlineMath","attrs":{"latex":"m = -2"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the point-slope form with the given point and new slope."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify to slope-intercept 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":"inlineMath","attrs":{"latex":"y = -2x + 7"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perpendicular lines have slopes that are negative reciprocals."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Write an equation of the line parallel to "},{"type":"inlineMath","attrs":{"latex":"x = -3"}},{"type":"text","marks":[{"type":"bold"}],"text":" through "},{"type":"inlineMath","attrs":{"latex":"(6, 2)"}}]},{"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: Vertical Lines"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of equations of vertical lines and their parallels."}]},{"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","text":"Vertical lines: "},{"type":"inlineMath","attrs":{"latex":"x = a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Parallel vertical lines have the same form but different "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-values."}]}]}]},{"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 line parallel to "},{"type":"inlineMath","attrs":{"latex":"x = -3"}},{"type":"text","text":" is also a vertical line."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The equation will be "},{"type":"inlineMath","attrs":{"latex":"x ="}},{"type":"text","text":" the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-coordinate of the given 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":"inlineMath","attrs":{"latex":"x = 6"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical lines parallel to "},{"type":"inlineMath","attrs":{"latex":"x = -3"}},{"type":"text","text":" have equations of the form "},{"type":"inlineMath","attrs":{"latex":"x = a"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. 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Find the midpoint between "},{"type":"inlineMath","attrs":{"latex":"(1, 9)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"(7, -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: Midpoint Formula"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the midpoint between 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","text":"Midpoint formula: "},{"type":"inlineMath","attrs":{"latex":"M = \\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)"}}]}]}]},{"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":"Add the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-coordinates and divide by 2."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-coordinates and divide by 2."}]}]}]},{"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":"inlineMath","attrs":{"latex":"(4, 3)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the midpoint formula to the given points."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q24. Find all points with "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-coordinate 2 that are 13 units from "},{"type":"inlineMath","attrs":{"latex":"(2, 5)"}}]},{"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 Formula (Vertical Distance)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the distance formula to find points a certain distance away, given a fixed "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-coordinate."}]},{"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","text":"Distance formula: "},{"type":"inlineMath","attrs":{"latex":"d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}"}}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" is fixed at 2, set up the distance formula with "},{"type":"inlineMath","attrs":{"latex":"x_2 = x_1 = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{(2 - 2)^2 + (y - 5)^2} = 13"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and solve 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":"inlineMath","attrs":{"latex":"(2, 18)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"(2, -8)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are two points, one above and one below the given point, both 13 units away."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks"