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

It represents two possible equations: x = a and x = -a, where a is a non-negative number.

Front

What is the solution to the equation |x| = -5?

Back

No solution, because absolute value cannot be negative.

Front

How do you solve the equation |x - 3| = 7?

Back

x - 3 = 7 or x - 3 = -7, leading to x = 10 or x = -4.

Front

What is the graphical representation of |x| < 3?

Back

The solution is the interval (-3, 3) on the number line.

Front

What does the inequality |x + 1| > 4 mean?

Back

It means x + 1 < -4 or x + 1 > 4, leading to x < -5 or x > 3.

Front

What is the solution to the inequality |x| ≤ 2?

Back

The solution is the interval [-2, 2] on the number line.

Front

What is the significance of the vertex in the graph of an absolute value function?

Back

The vertex represents the minimum or maximum point of the graph, where the function changes direction.

Front

What does the expression |x - 2| = 0 indicate?

Back

It indicates that x = 2 is the only solution, as absolute value equals zero only at that point.

Front

How do you interpret the graph of |x| > 1?

Back

The solution is x < -1 or x > 1, represented by two rays extending from -1 and 1.