SAT Math Practice Test: Radical, Rational and Absolute Value Equations
- Radical, Rational and Absolute Value Equations
- 10 questions
- 15 minutes
These 10 questions practice nonlinear equations from the SAT Advanced Math domain. After squaring both sides or clearing fractions, always check your answers in the original equation, because these steps can create extraneous solutions.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
If x + 5 = 4, what is the value of x?
Correct answer: C
Squaring both sides gives x + 5 = 16, so x = 11. The choice −1 comes from subtracting 5 from 4 without squaring.
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Question 2
What is the sum of the solutions to the equation |2x − 3| = 7?
Correct answer: B
The equation splits into 2x − 3 = 7 (so x = 5) and 2x − 3 = −7 (so x = −2). Their sum is 3. The choice 5 is only one of the two solutions.
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Question 3
What is the solution set of the equation 2x + 3 = x?
Correct answer: B
Squaring gives x² − 2x − 3 = 0, so x = 3 or x = −1. A square root cannot equal −1, so −1 is extraneous and only 3 works.
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Question 4
What is the solution to the equation 6x − 2 = 3x + 1?
Correct answer: A
Cross-multiplying gives 6(x + 1) = 3(x − 2), so 6x + 6 = 3x − 6 and x = −4. Check: 6/(−6) = −1 and 3/(−3) = −1.
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Question 5
If 1x + 12x = 14, what is the value of x?
Correct answer: C
The left side equals 3/(2x), so 3/(2x) = 1/4 and 2x = 12, giving x = 6. Check: 1/6 + 1/12 = 3/12 = 1/4.
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Question 6
Which inequality describes all solutions of |x − 4| < 3?
Correct answer: D
The distance from x to 4 must be less than 3, so −3 < x − 4 < 3, which gives 1 < x < 7. The "or" choice describes |x − 4| > 3 instead.
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Question 7
If x² − 9x − 3 = 10 and x ≠ 3, what is the value of x?
Correct answer: B
Since x² − 9 = (x − 3)(x + 3), the fraction simplifies to x + 3 = 10, so x = 7. The value 3 is excluded because it makes the denominator zero.
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Question 8
If ∛(x − 1) = −2, what is the value of x?
Correct answer: D
Cubing both sides gives x − 1 = (−2)³ = −8, so x = −7. The choice 5 comes from squaring −2 instead of cubing it.
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Question 9
The time T for one swing of a simple pendulum is given by T = 2πLg, where L is the length of the pendulum and g is a constant.
Which equation correctly expresses L in terms of T and g?
Correct answer: A
Divide by 2π to get T/(2π) = √(L/g), square to get T²/(4π²) = L/g, then multiply by g. The choice gT/(2π) skips the squaring step.
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Question 10
For x > 0, x · 4x = 18. What is the value of x?
Correct answer: D
The product equals √(4x²) = 2x, so 2x = 18 and x = 9. The choice 81 comes from treating the left side as √x = 9.
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