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 the equation |x| = a?

Back

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

Front

What does the graph of y = |x| look like?

Back

The graph of y = |x| is a V-shaped graph that opens upwards, with the vertex at the origin (0,0).

Front

What is the effect of a negative sign in front of the absolute value, such as in y = -|x|?

Back

The negative sign reflects the graph of |x| across the x-axis, creating an upside-down V shape.

Front

What does the inequality |x| < a represent?

Back

The inequality |x| < a represents all values of x that are within a distance a from 0 on the number line.

Front

What does the inequality |x| > a represent?

Back

The inequality |x| > a represents all values of x that are more than a distance a from 0 on the number line.

Front

How do you graph the inequality |x - 2| < 3?

Back

To graph |x - 2| < 3, find the critical points by solving x - 2 = 3 and x - 2 = -3, which gives x = 5 and x = -1. Shade the region between -1 and 5.

Front

What is the difference between using an open circle and a closed circle when graphing inequalities?

Back

An open circle indicates that the endpoint is not included (for < or >), while a closed circle indicates that the endpoint is included (for ≤ or ≥).

Front

How do you solve the absolute value inequality |x + 1| < 3?

Back

To solve |x + 1| < 3, set up the compound inequality: -3 < x + 1 < 3. Solve for x to get -4 < x < 2.

Front

What is the solution set for the equation |2x + 9| = 15?

Back

The solution set for |2x + 9| = 15 is x = 3 and x = -12.