Absolute Value Equations and 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 equation like |x| = a?

Back

To solve |x| = a, you set up two equations: x = a and x = -a.

Front

What does it mean if an absolute value inequality is written as |x| < a?

Back

It means that x is within a distance of a from zero, resulting in the compound inequality -a < x < a.

Front

What does it mean if an absolute value inequality is written as |x| > a?

Back

It means that x is more than a distance of a from zero, resulting in the compound inequalities x < -a or x > a.

Front

Solve the absolute value equation: |2x - 3| = 5.

Back

2x - 3 = 5 or 2x - 3 = -5; solutions are x = 4 and x = -1.

Front

Solve the inequality: |x + 2| ≤ 3.

Back

-3 ≤ x + 2 ≤ 3; solutions are -5 ≤ x ≤ 1.

Front

What is the solution to the inequality: |x| < 4?

Back

The solution is -4 < x < 4.

Front

What is the solution to the inequality: |x| > 2?

Back

The solution is x < -2 or x > 2.

Front

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

Back

Use a number line to indicate the range of values that satisfy the inequality, using open circles for < or > and closed circles for ≤ or ≥.

Front

What is the difference between an absolute value equation and an absolute value inequality?

Back

An absolute value equation has specific solutions, while an absolute value inequality describes a range of solutions.