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 represents the distance of a number from zero on the number line.

Front

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

Back

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

Front

What does the solution |x| ≤ a represent?

Back

The solution |x| ≤ a represents all values of x that are within a distance of a from zero, including the endpoints.

Front

What does the solution |x| ≥ a represent?

Back

The solution |x| ≥ a represents all values of x that are at least a distance of a from zero, including the endpoints.

Front

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

Back

To graph the solution, plot the critical points on a number line and shade the appropriate regions based on the inequality.

Front

What is the significance of the critical points in absolute value inequalities?

Back

Critical points are the values where the expression inside the absolute value equals zero or the boundary values of the inequality.

Front

How do you handle absolute value inequalities with negative values?

Back

When solving, ensure to consider the direction of the inequality and the implications of negative values on the solution set.

Front

What is the difference between strict and non-strict inequalities in absolute value?

Back

Strict inequalities (<, >) do not include the boundary points, while non-strict inequalities (≤, ≥) include the boundary points.

Front

How do you solve the inequality 3|2x + 7| - 7 < -4?

Back

First, isolate the absolute value: 3|2x + 7| < 3, then |2x + 7| < 1, leading to -1 < 2x + 7 < 1.