College Algebra Unit 1 Test – Step-by-Step Study Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Solve and check the linear equation: "},{"type":"inlineMath","attrs":{"latex":"(-7x + 3) + 5 = -6(x - 7)"}}]},{"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 in one variable and check your solution."}]},{"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 equation: An equation of the form "},{"type":"inlineMath","attrs":{"latex":"ax + b = c"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms: Add or subtract terms with the same variable."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate the variable: Get all terms with "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" on one side and constants on the other."}]}]}]},{"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":"Distribute the "},{"type":"inlineMath","attrs":{"latex":"-6"}},{"type":"text","text":" on the right side: "},{"type":"inlineMath","attrs":{"latex":"-6(x - 7)"}},{"type":"text","text":" becomes "},{"type":"inlineMath","attrs":{"latex":"-6x + 42"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms on the left: "},{"type":"inlineMath","attrs":{"latex":"(-7x + 3) + 5"}},{"type":"text","text":" becomes "},{"type":"inlineMath","attrs":{"latex":"-7x + 8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation: "},{"type":"inlineMath","attrs":{"latex":"-7x + 8 = -6x + 42"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add "},{"type":"inlineMath","attrs":{"latex":"7x"}},{"type":"text","text":" to both sides to get all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" terms on one side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $42$ from both sides to get all constants on the other side."}]}]}]},{"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 = -34"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After isolating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", you find "},{"type":"inlineMath","attrs":{"latex":"x = -34"}},{"type":"text","text":". Substitute back into the original equation to check your solution."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Solve the equation and state values that make denominators zero: "},{"type":"inlineMath","attrs":{"latex":"\\frac{4}{x+2} + \\frac{3}{x-2} = \\frac{12}{(x+2)(x-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: Rational Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving rational expressions and to identify restrictions on the variable."}]},{"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":"Rational equation: An equation involving fractions with polynomials in the numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Undefined values: Values that make any denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Common denominator: The least common multiple of all denominators."}]}]}]},{"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 values that make denominators zero: Set "},{"type":"inlineMath","attrs":{"latex":"x+2=0"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x-2=0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides of the equation by the least common denominator "},{"type":"inlineMath","attrs":{"latex":"(x+2)(x-2)"}},{"type":"text","text":" to clear fractions."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand and simplify each term after multiplying by the common denominator."}]}]},{"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":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check your solution(s) against the restrictions found in step 1 to ensure they do not make any denominator zero."}]}]}]},{"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 \\neq -2, 2;\\ \\varnothing"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The values "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" make denominators zero. After solving, no valid solution remains."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. The cost per ton of building an oil tanker is modeled by "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{119,000}{x} + 220"}},{"type":"text","marks":[{"type":"bold"}],"text":", where "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":" is the cost in dollars per ton and "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" is the tons (in thousands). What size of oil tanker (in thousands of tons) can be built for $200 per ton?"}]},{"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: Solving Rational Equations in Context"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply algebraic techniques to a real-world problem involving rational 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":"Model: An equation representing a real-world relationship."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute: Replace the variable "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" with the given value ($200$)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":": Rearranging the equation to isolate $x$."}]}]}]},{"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":"Substitute "},{"type":"inlineMath","attrs":{"latex":"y = 200"}},{"type":"text","text":" into the equation: "},{"type":"inlineMath","attrs":{"latex":"200 = \\frac{119,000}{x} + 220"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $220$ from both sides to isolate the rational term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation "},{"type":"inlineMath","attrs":{"latex":"200 - 220 = \\frac{119,000}{x}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by multiplying both sides by $x$ and then dividing both sides by the resulting coefficient."}]}]}]},{"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 = 375"}},{"type":"text","marks":[{"type":"bold"}],"text":" (thousand tons)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After solving, you find that a 375,000-ton oil tanker can be built for $200 per ton."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. When 5 times a number is subtracted from 7 times the number, the result is 16. What is the number?"}]},{"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: Translating and Solving Linear Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to translate a word problem into an algebraic equation and solve for the unknown."}]},{"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":"Let the unknown number be "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"\"5 times a number\" is "},{"type":"inlineMath","attrs":{"latex":"5x"}},{"type":"text","text":"; \"7 times a number\" is "},{"type":"inlineMath","attrs":{"latex":"7x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"\"Subtracted from\" means "},{"type":"inlineMath","attrs":{"latex":"7x - 5x"}},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" be the unknown number."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Translate the statement: "},{"type":"inlineMath","attrs":{"latex":"7x - 5x = 16"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms to simplify the equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"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 = 8"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The number is 8. This satisfies the original condition when checked."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. You inherit $42,000 to invest in two stocks paying 4% and 10% annual interest. How much should be invested at each rate if the total interest earned for the year is to be $2,400?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Systems of Linear Equations (Investment Problems)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve a system of equations based on an investment scenario."}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" = amount invested at 4%, "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" = amount invested at 10%."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total invested: "},{"type":"inlineMath","attrs":{"latex":"x + y = 42,000"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total interest: "},{"type":"inlineMath","attrs":{"latex":"0.04x + 0.10y = 2,400"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the two equations based on the information given."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve one equation for one variable (e.g., "},{"type":"inlineMath","attrs":{"latex":"y = 42,000 - x"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute this expression into the second equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and then use it to find "},{"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":"30,000"}},{"type":"text","marks":[{"type":"bold"}],"text":" at 4%; "},{"type":"inlineMath","attrs":{"latex":"12,000"}},{"type":"text","marks":[{"type":"bold"}],"text":" at 10%"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Invest "},{"type":"inlineMath","attrs":{"latex":"30,000"}},{"type":"text","text":" at 4% and "},{"type":"inlineMath","attrs":{"latex":"12,000"}},{"type":"text","text":" at 10% to earn "},{"type":"inlineMath","attrs":{"latex":"2,400"}},{"type":"text","text":" in interest."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. The length of a rectangular room is 9 feet longer than twice the width. If the room's perimeter is 198 feet, what are the room's dimensions?"}]},{"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: Perimeter and Linear Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve equations involving the perimeter of a rectangle."}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"w"}},{"type":"text","text":" = width, "},{"type":"inlineMath","attrs":{"latex":"l"}},{"type":"text","text":" = length."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Length: "},{"type":"inlineMath","attrs":{"latex":"l = 2w + 9"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perimeter of rectangle: "},{"type":"inlineMath","attrs":{"latex":"P = 2l + 2w"}},{"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":"Write the perimeter equation: "},{"type":"inlineMath","attrs":{"latex":"2l + 2w = 198"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"l = 2w + 9"}},{"type":"text","text":" into the perimeter equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand and combine like terms to form an equation in "},{"type":"inlineMath","attrs":{"latex":"w"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"w"}},{"type":"text","text":", then use it to find "},{"type":"inlineMath","attrs":{"latex":"l"}},{"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: Width = 30 ft; Length = 69 ft"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The room is 30 feet wide and 69 feet long."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. The sum of the angles of a triangle is 180°. Find the three angles if one angle is four times the smallest angle and the third angle is 30° greater than the smallest angle."}]},{"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: Triangle Angle Sum and Algebraic Expressions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve equations based on geometric relationships."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" = smallest angle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Second angle: "},{"type":"inlineMath","attrs":{"latex":"4x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Third angle: "},{"type":"inlineMath","attrs":{"latex":"x + 30"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum of angles: "},{"type":"inlineMath","attrs":{"latex":"x + 4x + (x + 30) = 180"}},{"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":"Write expressions for all three angles in terms of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation: "},{"type":"inlineMath","attrs":{"latex":"x + 4x + (x + 30) = 180"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms to simplify the equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", then use it to find the other two angles."}]}]}]},{"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: 25°, 100°, 55°"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The three angles are 25°, 100°, and 55°."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Solve the formula for "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","marks":[{"type":"bold"}],"text":": "},{"type":"inlineMath","attrs":{"latex":"F = \\frac{9}{5}C + 32"}}]},{"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: Solving Formulas for a Variable"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to rearrange a formula to solve for a specified variable."}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":" by performing inverse operations."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract 32, then multiply by the reciprocal of "},{"type":"inlineMath","attrs":{"latex":"\\frac{9}{5}"}},{"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":"Subtract 32 from both sides: "},{"type":"inlineMath","attrs":{"latex":"F - 32 = \\frac{9}{5}C"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides by "},{"type":"inlineMath","attrs":{"latex":"\\frac{5}{9}"}},{"type":"text","text":" to solve for "},{"type":"inlineMath","attrs":{"latex":"C"}},{"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":"C = \\frac{5}{9}(F - 32)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This formula converts Fahrenheit to Celsius."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. 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Solve the equation by the square root property: "},{"type":"inlineMath","attrs":{"latex":"(x - 7)^2 = -25"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations with Complex Solutions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve quadratic equations that have complex (imaginary) solutions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Square root property: "},{"type":"inlineMath","attrs":{"latex":"x^2 = a \\implies x = \\pm\\sqrt{a}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For negative "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{-a} = i\\sqrt{a}"}},{"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":"Take the square root of both sides, using "},{"type":"inlineMath","attrs":{"latex":"\\pm"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"i"}},{"type":"text","text":" for the negative under the root."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by adding 7 to both sides."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"x = 7 \\pm 5i"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The solutions are "},{"type":"inlineMath","attrs":{"latex":"x = 7 + 5i"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 7 - 5i"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. Solve the equation by the method of your choice: "},{"type":"inlineMath","attrs":{"latex":"5x^2 + 20x + 100 = 0"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations (Factoring, Quadratic Formula, or Completing the Square)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve a quadratic equation using an appropriate method."}]},{"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":"Quadratic formula: "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard form: "},{"type":"inlineMath","attrs":{"latex":"ax^2 + bx + c = 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":"Identify "},{"type":"inlineMath","attrs":{"latex":"a = 5"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"b = 20"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"c = 100"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug these values into the quadratic formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the discriminant "},{"type":"inlineMath","attrs":{"latex":"b^2 - 4ac"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Continue simplifying under the square root and set up the final expression 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 = -2 \\pm 4i"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The solutions are "},{"type":"inlineMath","attrs":{"latex":"x = -2 + 4i"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = -2 - 4i"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. The formula "},{"type":"inlineMath","attrs":{"latex":"P = 0.63x^2 - 0.048x + 3"}},{"type":"text","marks":[{"type":"bold"}],"text":" models the population "},{"type":"inlineMath","attrs":{"latex":"P"}},{"type":"text","marks":[{"type":"bold"}],"text":" (in thousands) for a species of fish "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" years after 1997. During what year will the population reach 65,520 fish?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations in Context"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve a quadratic equation in a real-world context."}]},{"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":"Set "},{"type":"inlineMath","attrs":{"latex":"P = 65.52"}},{"type":"text","text":" (since "},{"type":"inlineMath","attrs":{"latex":"P"}},{"type":"text","text":" is in thousands)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic equation: "},{"type":"inlineMath","attrs":{"latex":"ax^2 + bx + c = 0"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"0.63x^2 - 0.048x + 3 = 65.52"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract "},{"type":"inlineMath","attrs":{"latex":"65.52"}},{"type":"text","text":" from both sides to set the equation to zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the quadratic formula to solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add your solution for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to 1997 to find the year."}]}]}]},{"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: 2007"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The population reaches 65,520 fish in the year 2007."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Solve the radical equation and check all proposed solutions: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{24x + 48} = 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: Radical Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving square roots and to check for extraneous solutions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate the radical, then square both sides."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check all solutions in the original equation."}]}]}]},{"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":"Square both sides to eliminate the square root."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand and simplify the resulting quadratic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" using factoring or the quadratic formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check each solution in the original equation to ensure it is valid."}]}]}]},{"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 checking, only "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":" is a valid solution."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Solve and check the equation: "},{"type":"inlineMath","attrs":{"latex":"(x + 6)^{3/2} = 125"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Exponents and Solving Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations with rational exponents."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational exponent: "},{"type":"inlineMath","attrs":{"latex":"a^{m/n} = (\\sqrt[n]{a})^m"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate the base, then apply the inverse operation."}]}]}]},{"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 $125"},{"type":"inlineMath","attrs":{"latex":" as a power of $5"}},{"type":"text","text":" if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take both sides to the reciprocal power to isolate "},{"type":"inlineMath","attrs":{"latex":"x + 6"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by subtracting 6 from both sides."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"x = 19"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The solution is "},{"type":"inlineMath","attrs":{"latex":"x = 19"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Solve the equation by making an appropriate substitution: "},{"type":"inlineMath","attrs":{"latex":"x^4 - 2x^2 - 63 = 0"}}]},{"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: Solving Quartic Equations by Substitution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve higher-degree equations by substitution."}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x^2"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"x^4 = (x^2)^2 = u^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute and solve the resulting quadratic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Back-substitute to solve for "},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"u = x^2"}},{"type":"text","text":" and rewrite the equation in terms of "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the quadratic equation for "},{"type":"inlineMath","attrs":{"latex":"u"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"x^2 = u"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (consider both positive and negative roots, and complex solutions if necessary)."}]}]}]},{"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, 3, -i\\sqrt{7}, i\\sqrt{7}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The solutions are "},{"type":"inlineMath","attrs":{"latex":"x = -3, 3, -i\\sqrt{7}, i\\sqrt{7}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Solve the absolute value equation or indicate that the equation has no solution: "},{"type":"inlineMath","attrs":{"latex":"|3x - 2| + 6 = -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: Absolute Value Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the properties of absolute value and when equations have no solution."}]},{"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| \\geq 0"}},{"type":"text","text":" for all real "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If an absolute value expression equals a negative number, there is no 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":"Isolate the absolute value expression by subtracting 6 from both sides."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if the resulting equation is possible (can an absolute value equal a negative number?)."}]}]}]},{"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: No solution ("},{"type":"inlineMath","attrs":{"latex":"\\varnothing"}},{"type":"text","marks":[{"type":"bold"}],"text":")"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The equation has no solution because an absolute value cannot equal a negative number."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. The formula "},{"type":"inlineMath","attrs":{"latex":"v = \\sqrt{2gh}"}},{"type":"text","marks":[{"type":"bold"}],"text":" gives the velocity "},{"type":"inlineMath","attrs":{"latex":"v"}},{"type":"text","marks":[{"type":"bold"}],"text":" (in ft/s) of an object after falling height "},{"type":"inlineMath","attrs":{"latex":"h"}},{"type":"text","marks":[{"type":"bold"}],"text":" (in ft), with "},{"type":"inlineMath","attrs":{"latex":"g = 32"}},{"type":"text","marks":[{"type":"bold"}],"text":" ft/s"},{"type":"inlineMath","attrs":{"latex":"^2"}},{"type":"text","marks":[{"type":"bold"}],"text":". Find the velocity after falling 30 feet (round to the nearest tenth)."}]},{"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: Radical Equations in Physics Context"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to substitute values into a formula and evaluate a square root."}]},{"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":"inlineMath","attrs":{"latex":"v = \\sqrt{2gh}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"g = 32"}},{"type":"text","text":" ft/s"},{"type":"inlineMath","attrs":{"latex":"^2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"h = 30"}},{"type":"text","text":" ft"}]}]}]},{"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":"Substitute "},{"type":"inlineMath","attrs":{"latex":"g = 32"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"h = 30"}},{"type":"text","text":" into the formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"2gh"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the square root of the result."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Round your answer to the nearest tenth."}]}]}]},{"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":"v = 43.8"}},{"type":"text","marks":[{"type":"bold"}],"text":" ft/s"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The velocity after falling 30 feet is approximately 43.8 ft/s."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. Two cars leave an intersection, one north and one east. When the northbound car has gone 18 mi, the distance between the cars is 6 mi more than the distance traveled by the eastbound car. How far had the eastbound car traveled?"}]},{"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: Pythagorean Theorem in Word Problems"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply the Pythagorean theorem to a real-world scenario."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text"