Question 1

Solve the radical equation: $$x + \sqrt{x - 12} = 0. $$What is the solution?

Accepted Answer

$$x = 16$$

Question 2

Which of the following is the solution to the equation $$3\sqrt{x - 3} + 2 = 0$$?

Accepted Answer

$$x = 3$$

Question 3

Solve the equation $$2(\sqrt{x - 3})^2$ - 5\sqrt{x - 3} - 3 = 0. $$What is the solution?

Accepted Answer

$$x = 4 or$$ $$x = 9$$

Question 4

Solve and express in interval notation: $$-5 \leq 3x - 2 \u003c 13$$

Accepted Answer

$$[1, 5)$$

Question 5

Solve and express in interval notation: $$-4x - 17 \u003c 33$$

Accepted Answer

$$(-12.5, \infty)$$

Question 6

Solve and express in interval notation: $$3x + 7 \leq -1 or$$ $$x + 7 \u003e 12$$

Accepted Answer

$$(-\infty, -\dfrac{8}{3}] \cup (5, \infty)$$

Question 7

Solve the absolute value equation: $$|3x - 8| = 17. $$What are the solutions?

Accepted Answer

$$x = 9,\ x = -3$$

Question 8

Solve and express in interval notation: $$|2x + 1| \u003e 9$$

Accepted Answer

$$(-\infty, -5) \cup (4, \infty)$$

Question 9

Solve and express in interval notation: $$|5x - 4| + 2 \leq 18$$

Accepted Answer

$$[-2.8, 4.4]$$

Question 10

Solve and express in interval notation: $$2|x + 6| - 3 \u003c -7$$

Accepted Answer

$$(-\infty, -11) \cup (-1, \infty)$$

Question 11

A chemist is mixing two solutions and needs the temperature $$T (in$$ $$^\circC) to $$satisfy $$|2T - 10| \u003e 30$$ for a reaction to occur. What is the set of all possible temperatures?

Accepted Answer

$$(-\infty, -10) \cup (20, \infty)$$

Question 12

A company requires that the absolute difference between the number of products $$x$$ produced and 100 does not exceed 25. Which inequality represents this, and what is the solution in interval notation?

Accepted Answer

$$|x - 100| \leq 25$$, solution: $$[75, 125]$$

Question 13

Which of the following is the correct first step to solve the equation $$\sqrt{x + 5} = x - 1$$?

Accepted Answer

Isolate the radical and then square both sides

Question 14

A medical test result $$x is $$considered abnormal if $$|x - 98.6| \u003e 1.2. $$Which interval represents all abnormal results?

Accepted Answer

$$(-\infty, 97.4) \cup (99.8, \infty)$$

Question 15

Which of the following best describes the solution set for $$|x - 2| \u003c 5$$?

Accepted Answer

$$(-3, 7)$$

Question 16

A student solves $$|x + 4| = -2$$ and claims $$x = -6 is a $$solution. What is the error?

Accepted Answer

Absolute value equations cannot have negative solutions on the right side

Question 17

Which of the following is the correct solution to $$|2x - 3| \leq 7$$?

Accepted Answer

$$[-2, 5]$$

Question 18

If $$|x| \u003e 4$$, which of the following is the solution set?

Accepted Answer

$$(-\infty, -4) \cup (4, \infty)$$

Question 19

Which of the following best describes the solution to $$3x + 7 \leq -1$$?

Accepted Answer

$$x \leq -\dfrac{8}{3}$$

Precalculus Guidance: Radical, Absolute Value, and Inequality Equations

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