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 - 5| = 3?

Back

x - 5 = 3 or x - 5 = -3; thus, x = 8 or x = 2.

Front

What does the inequality |x| < a represent?

Back

It represents the values of x that are within a distance a from 0 on the number line.

Front

How do you solve the inequality |x + 2| > 4?

Back

x + 2 > 4 or x + 2 < -4; thus, x > 2 or x < -6.

Front

What is the graphical representation of |x| = a?

Back

It represents two points on the number line: x = a and x = -a.

Front

What does the expression |x - 3| + 2 = 5 imply?

Back

It implies that |x - 3| = 3, leading to x - 3 = 3 or x - 3 = -3; thus, x = 6 or x = 0.

Front

What is the effect of the negative sign in y = -|x|?

Back

It reflects the graph of |x| across the x-axis.

Front

How do you interpret the solution x < -3 or x > 12 in the context of | -2x + 9 | > 15?

Back

It indicates that x is either less than -3 or greater than 12, meaning the values of x are outside the interval [-3, 12].

Front

What is the solution set for |m - 2| < 8?

Back

The solution set is -6 < m < 10.

Front

What does the expression |x| ≥ a mean?

Back

It means that x is either greater than or equal to a or less than or equal to -a.