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?

Back

To solve an absolute value equation, set the expression inside the absolute value equal to both the positive and negative values of the other side of the equation.

Front

What does it mean if an absolute value inequality has no solution?

Back

It means that there are no values of the variable that can satisfy the inequality, often because the absolute value expression cannot be less than a negative number.

Front

Solve: |x - 3| = 5. What are the solutions?

Back

x = 8 or x = -2.

Front

What is the solution to the inequality |x + 4| < 3?

Back

-7 < x < -1.

Front

How do you graph an absolute value inequality?

Back

Graph the boundary line as a solid line for ≤ or ≥ and a dashed line for < or >, then shade the appropriate region based on the inequality.

Front

What is the difference between strict and non-strict inequalities?

Back

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

Front

Solve: |2x - 1| ≥ 3. What are the solutions?

Back

x ≤ -1 or x ≥ 2.

Front

What does the expression |x| > a imply?

Back

It implies that x is either greater than a or less than -a.

Front

How do you isolate the variable in an absolute value equation?

Back

First, remove the absolute value by setting up two separate equations, then solve each equation for the variable.