Solving Absolute-Value Inequalities

Front

What is an absolute value inequality?

Back

An inequality that involves the absolute value of a variable expression, indicating the distance of the expression from zero on a number line.

Front

How do you solve the inequality |x + 1| ≥ 3?

Back

x ≤ -4 or x ≥ 2.

Front

What does the solution x ≤ -4 or x ≥ 2 represent on a number line?

Back

It represents all values of x that are either less than or equal to -4 or greater than or equal to 2.

Front

Solve the inequality |3x + 6| ≥ 15.

Back

x ≤ -7 or x ≥ 3.

Front

What is the solution to the inequality |x + 3| > -8?

Back

All real numbers.

Front

Why does |x + 3| > -8 have all real numbers as a solution?

Back

Because the absolute value is always non-negative, so it is always greater than -8.

Front

Solve the inequality |2x + 13| < 15.

Back

-14 < x < 1.

Front

What does the solution -14 < x < 1 indicate?

Back

It indicates that x can take any value between -14 and 1, not including -14 and 1.

Front

How do you solve the inequality -5|x - 4| > -20?

Back

x < 8 and x > 0.

Front

What is the significance of the inequality -5|x - 4| > -20?

Back

It shows that the expression inside the absolute value must be less than 4 units away from 4.