Solving Absolute Value Inequalities

Front

What is an absolute value inequality?

Back

An absolute value inequality is an inequality that contains an absolute value expression, which measures the distance of a number from zero on the number line.

Front

How do you solve the inequality |x + 3| - 1 < 7?

Back

Add 1 to both sides: |x + 3| < 8. Then, split into two inequalities: -8 < x + 3 < 8. Solve to get -11 < x < 5.

Front

What does the solution -11 < x < 5 represent?

Back

It represents all real numbers x that are greater than -11 and less than 5.

Front

How do you solve the inequality -3|x - 3| > -24?

Back

Divide by -3 (remember to flip the inequality): |x - 3| < 8. Then, split into two inequalities: -8 < x - 3 < 8. Solve to get -5 < x < 11.

Front

What does the solution -5 < x < 11 represent?

Back

It represents all real numbers x that are greater than -5 and less than 11.

Front

How do you solve the inequality |2x + 1| ≤ 19?

Back

Split into two inequalities: -19 ≤ 2x + 1 ≤ 19. Solve to get -10 ≤ x ≤ 9.

Front

What does the solution -10 ≤ x ≤ 9 represent?

Back

It represents all real numbers x that are greater than or equal to -10 and less than or equal to 9.

Front

What is the first step in solving an absolute value inequality?

Back

Isolate the absolute value expression on one side of the inequality.

Front

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

Back

Plot the critical points on a number line and shade the region that satisfies the inequality.

Front

What is the solution to the inequality 4 + 2|x - 3| > 20?

Back

Subtract 4 from both sides: 2|x - 3| > 16. Divide by 2: |x - 3| > 8. Split into two inequalities: x - 3 > 8 or x - 3 < -8. Solve to get x > 11 or x < -5.