Absolute Value Inequalities

Front

What is the definition of absolute value?

Back

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.

Front

How do you solve an absolute value inequality of the form |x| < a?

Back

To solve |x| < a, split it into two inequalities: -a < x < a.

Front

How do you solve an absolute value inequality of the form |x| > a?

Back

To solve |x| > a, split it into two inequalities: x < -a or x > a.

Front

What does the inequality |x - c| < d represent?

Back

It represents the values of x that are within d units of c on the number line.

Front

What does the inequality |x - c| > d represent?

Back

It represents the values of x that are more than d units away from c on the number line.

Front

Solve: |2x + 3| < 5

Back

-4 < 2x + 3 < 4, which simplifies to -7 < 2x < 1, giving -3.5 < x < 0.5.

Front

Solve: |x - 1| ≥ 3

Back

x ≤ -2 or x ≥ 4.

Front

What is the solution set for |x + 2| < 6?

Back

-8 < x < 4.

Front

What is the solution set for |x - 5| > 2?

Back

x < 3 or x > 7.

Front

How do you graph the solution of an absolute value inequality?

Back

Use open circles for < and >, and closed circles for ≤ and ≥, then shade the appropriate regions.