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

What does the equation |x| = a represent?

Back

The equation |x| = a has two solutions: x = a and x = -a, where a is a non-negative number.

Front

What does the inequality |x| > a represent?

Back

The inequality |x| > a means that x is either less than -a or greater than a, represented as x < -a or x > a.

Front

What does the inequality |x| < a represent?

Back

The inequality |x| < a means that x is between -a and a, represented as -a < x < a.

Front

What is the solution set for |x| ≤ a?

Back

The solution set for |x| ≤ a is the closed interval [-a, a].

Front

How do you solve the equation |x + 4| = 7?

Back

To solve |x + 4| = 7, set up two equations: x + 4 = 7 and x + 4 = -7. The solutions are x = 3 and x = -11.

Front

What is the solution set for |x| > 15?

Back

The solution set for |x| > 15 is (-∞, -15) ∪ (15, ∞).

Front

How do you solve the inequality |x + 3| < 5?

Back

To solve |x + 3| < 5, set up the compound inequality -5 < x + 3 < 5. This simplifies to -8 < x < 2.

Front

What is the graphical representation of |x| = 15?

Back

The graphical representation of |x| = 15 consists of two points on the number line: -15 and 15.

Front

What does the inequality |x| ≤ 15 indicate about x?

Back

The inequality |x| ≤ 15 indicates that x is within the range of -15 to 15, inclusive.