Algebra 2 Practice Test: Solving Rational Equations
- Solving Rational Equations
- 10 questions
- 15 minutes
These 10 questions cover equations with variables in the denominator. Tip: multiply by the least common denominator, then check every answer in the original equation, because some may make a denominator zero.
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
Solve 3x + 12 = 2.
Correct answer: A
Subtract 12: 3x = 32, so x = 2. Check: 32 + 12 = 2.
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Question 2
Solve xx − 3 = 3x − 3 + 2.
Correct answer: D
Multiplying by x − 3 gives x = 3 + 2(x − 3), so x = 3. But x = 3 makes the denominators zero, so it is extraneous and the equation has no solution.
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Question 3
Solve x + 14 = 5x.
Correct answer: B
Cross-multiply: x(x + 1) = 20, so x2 + x − 20 = 0 and (x + 5)(x − 4) = 0. Neither 4 nor −5 makes a denominator zero, so both are solutions.
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Question 4
What is the least common denominator of 1x + 2 and 3x2 − 4?
Correct answer: C
x2 − 4 = (x + 2)(x − 2) already contains the factor x + 2, so the LCD is (x + 2)(x − 2). Multiplying the two denominators gives a common denominator, but not the least one.
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Question 5
Solve 2x − 1 = 6x + 3.
Correct answer: B
Cross-multiply: 2(x + 3) = 6(x − 1), so 2x + 6 = 6x − 6 and x = 3. Check: 22 = 1 and 66 = 1.
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Question 6
One pipe can fill a tank in 6 hours. A second pipe can fill it in 3 hours. How long does it take both pipes together?
Correct answer: D
Rates add: 16 + 13 = 12 tank per hour, so together they need 2 hours. Averaging 6 and 3 to get 4.5 hours ignores that the pipes work at the same time.
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Question 7
Lena can paint a fence alone in 4 hours. Working together with Omar, they paint it in 3 hours. How long would Omar take alone?
Correct answer: A
14 + 1t = 13, so 1t = 13 − 14 = 112 and t = 12. The answer 1 hour comes from subtracting the times instead of the rates.
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Question 8
A boat travels 24 miles downstream in the same time it travels 16 miles upstream. The current flows at 2 mph. What is the speed of the boat in still water?
Correct answer: C
Time = distance ÷ speed: 24b + 2 = 16b − 2. Then 24b − 48 = 16b + 32, so b = 10. Check: 24 ÷ 12 = 2 hours and 16 ÷ 8 = 2 hours.
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Question 9
Solve x2x − 4 = 16x − 4.
Correct answer: D
Multiplying by x − 4 gives x2 = 16, so x = ±4. The value 4 makes the denominator zero and is extraneous, so only x = −4 works.
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Question 10
Solve 1x + 12x = 14.
Correct answer: C
Write the left side over 2x: 22x + 12x = 32x. Then 32x = 14, so 2x = 12 and x = 6. Stopping at 2x = 12 gives the trap 12.
Questions you leave blank are not counted as wrong.
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