Absolute Value Equations and Inequalities Test

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 + 10| = 11?

Back

To solve |x + 10| = 11, set up two equations: x + 10 = 11 and x + 10 = -11. Solving these gives x = 1 and x = -21.

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 solution set for the inequality |x| < 3?

Back

The solution set for |x| < 3 is -3 < x < 3.

Front

How do you interpret the inequality |x - 4| > 2?

Back

The inequality |x - 4| > 2 means that x is either less than 2 or greater than 6.

Front

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

Back

A negative coefficient reflects the graph of the absolute value function across the x-axis.

Front

How do you graph the inequality |x + 2| ≤ 4?

Back

To graph |x + 2| ≤ 4, find the critical points by solving x + 2 = 4 and x + 2 = -4, which gives x = 2 and x = -6. Shade the region between -6 and 2, including the endpoints.

Front

What is the solution to the equation 9 - 2|4x - 2| = -3?

Back

To solve 9 - 2|4x - 2| = -3, first isolate the absolute value: 2|4x - 2| = 12, then |4x - 2| = 6. This gives two equations: 4x - 2 = 6 and 4x - 2 = -6, leading to x = 2 and x = -1.

Front

What does the graph of y = -2|x + 4| represent?

Back

The graph of y = -2|x + 4| is a V-shaped graph that opens downwards, with the vertex at (-4, 0).

Front

What is the relationship between absolute value equations and their corresponding inequalities?

Back

Absolute value equations can be transformed into inequalities by considering the distance from a point, leading to two cases for equations and a range of solutions for inequalities.