Absolute Value Equations and Inequalities Worksheet

Question 1

Solve the equation |x| = 7.

Accepted Answer

The solutions are x = 7 and x = -7 because both 7 and -7 are 7 units away from 0.

Question 2

Solve the equation |x - 3| = 5.

Accepted Answer

The solutions are x = 8 and x = -2 because x - 3 = 5 or x - 3 = -5.

Question 3

Solve the equation |2x| = 12.

Accepted Answer

The solutions are x = 6 and x = -6 because 2x = 12 or 2x = -12, and dividing by 2 gives x = 6 or x = -6.

Question 4

Solve the equation |x + 4| = 9.

Accepted Answer

The solutions are x = 5 and x = -13 because x + 4 = 9 or x + 4 = -9.

Question 5

Solve the equation |3x - 1| = 8.

Accepted Answer

The solutions are x = 3 and x = -7/3 because 3x - 1 = 8 or 3x - 1 = -8. Solving gives 3x = 9 so x = 3, or 3x = -7 so x = -7/3.

Question 6

Does the equation |x - 2| = -4 have a solution? Explain.

Accepted Answer

This equation has no solution because an absolute value can never be negative. The value of |x - 2| is always 0 or greater.

Question 7

Solve the inequality |x| < 6.

Accepted Answer

The solution is -6 < x < 6 because all numbers less than 6 units from 0 are between -6 and 6.

Question 8

Solve the inequality |x| > 4.

Accepted Answer

The solution is x < -4 or x > 4 because numbers more than 4 units from 0 lie to the left of -4 or to the right of 4.

Question 9

Solve the inequality |x + 1| <= 3.

Accepted Answer

The solution is -4 <= x <= 2 because x + 1 must be between -3 and 3. Subtracting 1 gives -4 <= x <= 2.

Question 10

Solve the inequality |x - 5| >= 2.

Accepted Answer

The solution is x <= 3 or x >= 7 because x - 5 <= -2 or x - 5 >= 2. Adding 5 gives x <= 3 or x >= 7.

Question 11

Solve the inequality |2x - 6| < 10.

Accepted Answer

The solution is -2 < x < 8 because -10 < 2x - 6 < 10. Adding 6 gives -4 < 2x < 16, and dividing by 2 gives -2 < x < 8.

Question 12

Solve the inequality |3x + 3| > 9.

Accepted Answer

The solution is x < -4 or x > 2 because 3x + 3 < -9 or 3x + 3 > 9. Solving gives 3x < -12 so x < -4, or 3x > 6 so x > 2.

Question 13

Write an absolute value inequality for this statement: The distance between x and 10 is at most 4.

Accepted Answer

The inequality is |x - 10| <= 4 because absolute value represents distance from 10, and at most 4 means less than or equal to 4.

Question 14

A temperature is within 3 degrees of 20 degrees. Write and solve an absolute value inequality.

Accepted Answer

One correct inequality is |t - 20| <= 3. The solution is 17 <= t <= 23 because the temperature can be at most 3 degrees above or below 20.

Question 15

A student solved |x - 4| = 6 and got x = 10 only. Find the missing solution and explain the error.

Accepted Answer

The missing solution is x = -2. The error is that the student solved only x - 4 = 6 and forgot to also solve x - 4 = -6.

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