Absolute Value 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 inequality |x| < a represent?

Back

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

Front

What does the inequality |x| > a represent?

Back

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

Front

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

Back

Split into two inequalities: -3 < x - 5 < 3, which simplifies to 2 < x < 8.

Front

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

Back

Split into two inequalities: x + 2 < -4 or x + 2 > 4, which simplifies to x < -6 or x > 2.

Front

What is the solution to the inequality |2x - 1| ≤ 3?

Back

Split into two inequalities: -3 ≤ 2x - 1 ≤ 3, which simplifies to -1 ≤ x ≤ 2.

Front

What is the graphical representation of |x| < 2?

Back

It is represented by an open interval (-2, 2) on the number line.

Front

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

Back

It is represented by two rays: x ≤ -3 and x ≥ 3 on the number line.

Front

How do you interpret the solution x < -3 or x > 2?

Back

It means that x is either less than -3 or greater than 2, indicating two separate ranges.

Front

What is the solution to the inequality 3|x + 1| < 9?

Back

Divide by 3: |x + 1| < 3, which gives -3 < x + 1 < 3, simplifying to -4 < x < 2.