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, isolate the absolute value expression and then set up two separate equations: one for the positive case and one for the negative case.

Front

What does the inequality |x| < a represent?

Back

The inequality |x| < a means that x is within a distance of a from zero, resulting in the solution -a < x < a.

Front

What does the inequality |x| > a represent?

Back

The inequality |x| > a means that x is more than a distance of a from zero, resulting in the solution x < -a or x > a.

Front

What is the first step in solving the equation 4|5x-2|+7>39?

Back

The first step is to isolate the absolute value by subtracting 7 from both sides, resulting in 4|5x-2| > 32.

Front

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

Back

To graph the solution of an absolute value inequality, plot the critical points on a number line and use open or closed circles based on whether the inequality is strict (<, >) or inclusive (≤, ≥).

Front

What is the solution set for the equation −2|−2r − 4| = -12?

Back

The solution set is {-5, 1}.

Front

What does it mean to split an absolute value equation into two equations?

Back

Splitting an absolute value equation into two equations means creating one equation for the positive case and another for the negative case of the expression inside the absolute value.

Front

What is the correct way to express the solution x = -2 and x = 6?

Back

The correct way to express this solution is x = -2 OR x = 6.

Front

What is the significance of the absolute value symbol in equations?

Back

The absolute value symbol indicates that the expression can take on both positive and negative values, leading to multiple possible solutions.