Question 1

Which of the following is the correct first step to solve the rational equation $$\displaystyle \frac{x-8}{3} + \frac{x-3}{2} = 0$$?

Accepted Answer

Multiply both sides by the least common denominator, which is $$6.$$

Question 2

After multiplying both sides of $$\displaystyle \frac{x-8}{3} + \frac{x-3}{2} = 0 by $$the LCD, what is the resulting equation?

Accepted Answer

$$2(x-8) + 3(x-3) = 0$$

Question 3

What is the solution to the rational equation $$\displaystyle \frac{x^2$}{x-3} = \frac{9}{x-3}$$?

Accepted Answer

$$x = -3$$ only

Question 4

Why is $$x = 3$$ not a valid solution for $$\displaystyle \frac{x^2$}{x-3} = \frac{9}{x-3}$$?

Accepted Answer

Because it results in division by zero.

Question 5

Which principle is used to eliminate radicals when solving equations such as $$\sqrt{3x+1} = 4$$?

Accepted Answer

Principle of Powers

Question 6

What is the solution to the radical equation $$\sqrt{3x+1} = 4$$?

Accepted Answer

$$x = 5$$

Question 7

When solving $$5 + \sqrt{x+7} = x$$, what is the first step?

Accepted Answer

Isolate $$\sqrt{x+7} on $$one side.

Question 8

After isolating the radical in $$5 + \sqrt{x+7} = x$$, what equation do you get after squaring both sides?

Accepted Answer

$$x+7 = (x-5)^2$

Question 9

What are the possible solutions to $$5 + \sqrt{x+7} = x$$ before checking for extraneous solutions?

Accepted Answer

$$x = 9$$ and $$x = 2$$

Question 10

After checking, which value is the actual solution to $$5 + \sqrt{x+7} = x$$?

Accepted Answer

$$x = 9$$

Question 11

Which of the following is a necessary step when solving a radical equation with two radical terms, such as $$\sqrt{2x-5} - \sqrt{x-3} = 1$$?

Accepted Answer

Isolate one radical and apply the Principle of Powers.

Question 12

A patient’s medication dosage is modeled by the equation $$\sqrt{3x+1} = 4$$, where $$x is $$the number of hours since administration. How many hours until the medication reaches the required level?

Accepted Answer

$$x = 5$$

Question 13

Which of the following best describes an extraneous solution in the context of rational and radical equations?

Accepted Answer

A solution that does not satisfy the original equation after substitution.

Question 14

A chemical reaction rate is modeled by $$\sqrt{x+5} = 3. $$What is the value of $$x$$ when the rate reaches $$3$$?

Accepted Answer

$$x = 4$$

Question 15

Which of the following is the correct way to check for extraneous solutions after solving a radical equation?

Accepted Answer

Substitute each solution back into the original equation.

Question 16

Solve the rational equation $$\displaystyle \frac{2x}{x-1} = 4. $$What is the solution?

Accepted Answer

$$x = \frac{4}{3}$$

Question 17

A biologist models population growth with $$\sqrt{2x-1} = 5. $$What is the value of $$x$$ when the population reaches the modeled level?

Accepted Answer

$$x = 12$$

Question 18

Which of the following is NOT a valid step when solving a rational equation?

Accepted Answer

Dividing both sides by zero.

Question 19

Solve the radical equation $$\sqrt{x+2} + 2 = 6. $$What is the solution?

Accepted Answer

$$x = 12$$

Solving Rational and Radical Equations (Section 3.4) – Precalculus Study Notes

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