Absolute Value Equations, Inequalities, and Graphing Review

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 - 3| = 5?

Back

To solve |x - 3| = 5, set up two equations: x - 3 = 5 and x - 3 = -5. The solutions are x = 8 and x = -2.

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

How do you graph the equation y = |x - 1|?

Back

To graph y = |x - 1|, shift the graph of y = |x| to the right by 1 unit. The vertex is at (1,0).

Front

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

Back

The solution set is -7 < x < -1.

Front

How do you solve the inequality 4|5x - 2| + 7 > 39?

Back

First, isolate the absolute value: 4|5x - 2| > 32. Then, divide by 4: |5x - 2| > 8. This leads to two inequalities: 5x - 2 > 8 or 5x - 2 < -8.

Front

What are the critical points when solving |2a - 10| = 6?

Back

Set up two equations: 2a - 10 = 6 and 2a - 10 = -6. The critical points are a = 8 and a = 2.

Front

What does the expression |v + 8| - 5 = 2 represent?

Back

This represents an absolute value equation that can be solved by isolating the absolute value and setting up two cases.

Front

How do you interpret the solution set {-1, -15}?

Back

The solution set {-1, -15} indicates that both -1 and -15 satisfy the original absolute value equation.

Front

What is the vertex of the graph of y = |x - 3| - 1?

Back

The vertex of the graph is at the point (3, -1).