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 the inequality |x| < a represent?

Back

It represents that x is within a distance a from 0, meaning -a < x < a.

Front

What does the inequality |x| > a represent?

Back

It represents that x is outside a distance a from 0, meaning x < -a or x > a.

Front

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

Back

Graph the critical points found from the inequality and use open or closed circles based on whether the inequality is strict (<, >) or inclusive (≤, ≥).

Front

What is the solution to the equation |x - 3| = 5?

Back

x - 3 = 5 or x - 3 = -5, leading to x = 8 or x = -2.

Front

What is the solution to the inequality |2x + 1| ≤ 3?

Back

-4 ≤ 2x ≤ 2, leading to -2 ≤ x ≤ 1.

Front

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

Back

An absolute value equation has an equal sign, while an absolute value inequality has an inequality sign (>, <, ≥, ≤).

Front

What does the term 'no solution' mean in the context of inequalities?

Back

It means that there are no values of the variable that satisfy the inequality.

Front

What does 'infinite solutions' mean in the context of inequalities?

Back

It means that any value of the variable satisfies the inequality.