Absolute Value Equations and Inequalities 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 an absolute value equation of the form |x| = a?

Back

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

Front

What does the inequality |x| < a represent?

Back

The inequality |x| < a represents the set of all points whose distance from zero is less than a, which can be expressed as -a < x < a.

Front

How do you solve an absolute value inequality of the form |x| > a?

Back

To solve |x| > a, you set up two inequalities: x < -a or x > a.

Front

What is the graphical representation of |x - c| = d?

Back

The graphical representation of |x - c| = d consists of two points on the number line: c + d and c - d.

Front

What does the expression |x + 12| < 2 imply about x?

Back

The expression |x + 12| < 2 implies that x is within 2 units of -12, leading to the inequality -14 < x < -10.

Front

How do you interpret the solution x < 8 and x > 0 in the context of absolute value inequalities?

Back

The solution x < 8 and x > 0 indicates that x can take any value between 0 and 8, not including 0 and 8.

Front

What is the significance of the distance in the context of absolute value equations?

Back

In absolute value equations, the distance represents how far a number is from a specific point on the number line.

Front

How do you express the solution to |x - 4| = 3 in terms of distance?

Back

The solution |x - 4| = 3 can be expressed as the set of points whose distance from 4 is equal to 3.

Front

What is the first step in solving the inequality -5|x - 4| > -20?

Back

The first step is to divide both sides by -5, remembering to reverse the inequality sign, resulting in |x - 4| < 4.