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Radical and rational equations — ACT practice questions

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• 45 questions on the paper
• 38 practice questions in the app
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Practice 38 Radical and rational equations questions in the app

What this topic is

Radical and rational equations on the ACT Mathematics test involve solving for a variable under a square root or in a denominator, and translating combined-work stories into rate equations.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

What is the solution of $\dfrac{3}{x}+\dfrac{1}{2}=\dfrac{5}{x}$?

A0✗ Not this oneB2✗ Not this oneC4✓ CorrectD6✗ Not this one

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Answer: C — 4

Subtracting $\dfrac{3}{x}$ from both sides gives $\dfrac{2}{x}=\dfrac{1}{2}$, so $x=4$. The value 0 is excluded because it makes a denominator zero.

Question 2Mid

One pump fills a tank in $x$ hours, and another pump fills the same tank in $x+6$ hours. Working together, the two pumps fill the tank in 4 hours. What is the value of $x$?

A4✗ Not this oneB6✓ CorrectC10✗ Not this oneD12✗ Not this one

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Answer: B — 6

Clearing denominators gives $x^2-2x-24=0$, so the candidates are 6 and $-4$. Because a completion time is positive, $x=6$.

Question 3Mid

What is the solution of $\sqrt{2x+15}=x+3$?

A$-2-\sqrt{10}$✗ Not this oneB$2-\sqrt{10}$✗ Not this oneC$-2+\sqrt{10}$✓ CorrectD$2+\sqrt{10}$✗ Not this one

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Answer: C — $-2+\sqrt{10}$

Squaring gives $x^2+4x-6=0$, whose roots are $-2\pm\sqrt{10}$. The smaller root makes $x+3$ negative, so only $-2+\sqrt{10}$ works.

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