Solve Absolute Value Equations & Inequalities

Front

What is an 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, you set up two separate equations: one for the positive case and one for the negative case, then solve each equation.

Front

What is the solution set for the equation |x| = a, where a > 0?

Back

The solution set is x = a or x = -a.

Front

What does the inequality |x| < a represent?

Back

The inequality |x| < a represents the set of all x values that are within a distance of a from zero, which can be expressed as -a < x < a.

Front

What does the inequality |x| > a represent?

Back

The inequality |x| > a represents the set of all x values that are more than a distance of a from zero, which can be expressed as x < -a or x > a.

Front

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

Back

To graph the solution of an absolute value inequality, first solve the inequality, then plot the critical points on a number line and shade the appropriate regions.

Front

What is the first step in solving the equation |x + 3| = 7?

Back

The first step is to set up two equations: x + 3 = 7 and x + 3 = -7.

Front

What is the solution to the equation |x - 4| = 2?

Back

The solutions are x = 6 and x = 2.

Front

How do you interpret the solution set of an absolute value inequality?

Back

The solution set indicates the range of values that satisfy the inequality, which can be expressed in interval notation.

Front

What is the difference between strict and non-strict inequalities in the context of absolute values?

Back

Strict inequalities (e.g., |x| < a) do not include the endpoints, while non-strict inequalities (e.g., |x| ≤ a) include the endpoints.