Question 1

Which of the following is the correct first step to solve the rational equation $$\displaystyle \frac{2x+1}{x-3} = \frac{4}{x}$$?

Accepted Answer

Multiply both sides by $$x(x-3)$$

Question 2

Given the equation $$\displaystyle \frac{3x+2}{x-2} + \frac{1}{x} = \frac{-2}{x^2$-2x}$$, which values of $$x$$ must be excluded from the solution set?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 3

A plumber can fix a leak in 4 hours, while an apprentice takes 6 hours. Working together, how long will it take them to fix the leak?

Accepted Answer

$$2.4$$ hours

Question 4

Solve for $$x$$: $$\displaystyle \frac{5x-2}{x+1} = 3$$

Accepted Answer

$$x = \frac{3}{2}$$

Question 5

If $$\sqrt{x+5} = x-1$$, what is the solution for $$x$$?

Accepted Answer

$$x = 2$$

Question 6

Which of the following best describes the power property as used in solving equations?

Accepted Answer

If $$P = Q$$, then every solution of $$P = Q is $$also a solution of $$P^n = Q^n$$ for any positive integer $$n$$

Question 7

Solve for $$x$$: $$\sqrt{2x+3} = x$$

Accepted Answer

$$x = 3$$

Question 8

Which of the following is a valid step when solving an equation involving a radical, such as $$\sqrt{x+2} = 4$$?

Accepted Answer

Square both sides to eliminate the radical

Question 9

A painter can paint a house in 8 hours. If she works with a helper and together they finish in 5 hours, how long would it take the helper alone?

Accepted Answer

$$13.3$$ hours

Question 10

Which equation is quadratic in form?

Accepted Answer

$$x^4$$ + $$2x^2 - 3 = 0$

Question 11

Solve for $$x$$: $$(x+1)^{2/3}$$ - $$(x+1)^{1/3} - 2 = 0$

Accepted Answer

$$x = 7, x = -2$$

Question 12

Which of the following is NOT a valid solution to $$\frac{3x-1}{3} - \frac{2x}{x-1} = x$$?

Accepted Answer

$$x = 1$$

Question 13

If $$\frac{a}{b} + \frac{c}{d} = 1$$ and $$b = d$$, what is the value of $$a + c$$?

Accepted Answer

$$b$$

Question 14

Which step is essential when solving equations involving radicals to avoid extraneous solutions?

Accepted Answer

Check each proposed solution in the original equation

Question 15

If a machine can complete a task in $$t$$ hours, what is its rate of work?

Accepted Answer

$$\frac{1}{t}$$ jobs per hour

Question 16

Solve for $$x$$: $$\sqrt{4x^2$ - 4x + 1} = x$$

Accepted Answer

$$x = 1$$

Question 17

Which of the following best describes a rational equation?

Accepted Answer

An equation with at least one rational expression

Question 18

If $$u = (x+1$$)^{1/3}$$ and u^2$ - u - 2 = 0$$, what are the possible values for $$u$$?

Accepted Answer

$$u = 2$$ and $$u = -1$$

Question 19

Which of the following is the correct solution set for $$4x^2 - 5x + 1 = 0$?

Accepted Answer

$$x = 1, x = \frac{1}{4}$$

Rational Equations and Related Concepts in College Algebra

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