{"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 describe the set of solutions 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 flips."}]}]}]},{"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 expressed the solution as an interval."}]}]},{"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 asks you to solve a linear inequality and find the solution set."}]},{"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":"The solution in interval notation is "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, 13)"}},{"type":"text","text":"."}]}]},{"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 on a number line."}]},{"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 a fraction, 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 and appropriate symbols (closed or open circles)."}]}]}]},{"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 to graph it."}]}]}]},{"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":"Interval notation: "},{"type":"inlineMath","attrs":{"latex":"[9, \\infty)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"On a number line, this is a closed circle at 9 and shading to the right."}]}]},{"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 problem involves solving a linear inequality with a negative coefficient and applying the distributive property."}]},{"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":"When dividing or multiplying both sides of an inequality by a negative number, reverse the inequality sign."}]}]}]},{"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":"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 $x$. Remember to flip 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":"Interval notation: "},{"type":"inlineMath","attrs":{"latex":"[-2, \\infty)"}}]}]},{"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 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","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 inside."}]}]},{"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 values, specifically the 'less than' case."}]},{"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":", the solution is "},{"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":"Expressed in interval notation: "},{"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 involves solving an absolute value inequality with a negative coefficient."}]},{"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":", the solution is "},{"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 flip the inequality sign."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the resulting inequality in standard absolute value form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the compound inequality 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":"-1 \\leq x \\leq 9"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interval notation: "},{"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 a linear equation for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"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","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combining like 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":"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 that involve 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 of 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":"Combine like terms 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 fractions 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 involves solving a linear equation with the distributive property."}]},{"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":"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 "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" terms 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, you find the value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"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: Slope-Intercept 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":"The slope is 5 and the y-intercept is -10."}]}]},{"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 asks you 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":"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 "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the values of "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" (slope) and "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" (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: "},{"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 slope is -2 and the y-intercept is -3."}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"a = 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":"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 an undefined slope and do not cross the y-axis (no 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 have a y-intercept."}]}]},{"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 through 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, if required."}]}]}]},{"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":"The line through (1, -2) and (5, 6) has slope 2 and y-intercept -4."}]}]},{"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: Point-Slope and Slope-Intercept Form"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to write the equation of a line given a point and a slope."}]},{"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 point and slope 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":"The equation passes through (2, 3) with slope -4."}]}]},{"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 (-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 understanding of parallel lines and how to write their equations."}]},{"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":"The parallel line has the same slope and passes through (-1, 4)."}]}]},{"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 (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 find 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: if "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"-1/m"}},{"type":"text","text":"."}]}]},{"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 slope of the given line ("},{"type":"inlineMath","attrs":{"latex":"m = 1/2"}},{"type":"text","text":"), so the perpendicular slope is "},{"type":"inlineMath","attrs":{"latex":"-2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the point-slope form with the perpendicular slope and the given point."}]}]},{"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":"The perpendicular line has slope -2 and passes through (4, -1)."}]}]},{"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 (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 vertical lines and their equations."}]},{"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":"a"}},{"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":"The parallel vertical line passes through (6, 2)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Write an equation of the line perpendicular to "},{"type":"inlineMath","attrs":{"latex":"y = -6"}},{"type":"text","marks":[{"type":"bold"}],"text":" through (0, 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: Perpendicular Lines (Horizontal and Vertical)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between horizontal and 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":"Horizontal line: "},{"type":"inlineMath","attrs":{"latex":"y = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perpendicular to a horizontal line is a vertical line: "},{"type":"inlineMath","attrs":{"latex":"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 the perpendicular to "},{"type":"inlineMath","attrs":{"latex":"y = -6"}},{"type":"text","text":" is 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 = 0"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The perpendicular line passes through (0, 5)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. Find the distance between (-2, 7) and (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: Distance Formula"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the distance formula to find the distance between two points in the plane."}]},{"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":"Label the points: "},{"type":"inlineMath","attrs":{"latex":"(x_1, y_1) = (-2, 7)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(x_2, y_2) = (4, -1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"x_2 - x_1"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y_2 - y_1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Square each difference and add them together."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the square root of the sum to find the distance."}]}]}]},{"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":"6\\sqrt{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The distance between the points is "},{"type":"inlineMath","attrs":{"latex":"6\\sqrt{2}"}},{"type":"text","text":" units."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. Find the distance between (5, 3) and (5, -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: Distance Formula"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the distance formula, especially when the points have the same "},{"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":"Label the points: "},{"type":"inlineMath","attrs":{"latex":"(x_1, y_1) = (5, 3)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(x_2, y_2) = (5, -9)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"x_2 - x_1"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y_2 - y_1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Since "},{"type":"inlineMath","attrs":{"latex":"x_2 - x_1 = 0"}},{"type":"text","text":", focus on the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the absolute value of the difference in "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-coordinates."}]}]}]},{"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: $12$"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The distance is the absolute difference in "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-coordinates."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q22. Find the midpoint of the line segment with endpoints (-6, 4) and (8, -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: 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":"Label the points: "},{"type":"inlineMath","attrs":{"latex":"(x_1, y_1) = (-6, 4)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(x_2, y_2) = (8, -2)"}},{"type":"text","text":"."}]}]},{"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":"(1, 1)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The midpoint is at (1, 1)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q23. Find the midpoint between (1, 9) and (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 use the midpoint formula."}]},{"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":"Label the points: "},{"type":"inlineMath","attrs":{"latex":"(x_1, y_1) = (1, 9)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(x_2, y_2) = (7, -3)"}},{"type":"text","text":"."}]}]},{"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":"The midpoint is at (4, 3)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q24. Find all points with x-coordinate 2 that are 13 units from (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"}]},{"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":"Let the unknown point be "},{"type":"inlineMath","attrs":{"latex":"(2, y)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the distance formula: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{(2 - 2)^2 + (y - 5)^2} = 13"}},{"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 with "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" that are 13 units from (2, 5)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q25. Find all points with y-coordinate -4 that are 10 units from (6, -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: Distance Formula"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the distance formula with a fixed "},{"type":"inlineMath","attrs":{"latex":"y"}},{"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":"pa