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 - 4| < 5?

Back

Split into two inequalities: -5 < x - 4 < 5, which simplifies to -1 < x < 9.

Front

How do you solve the inequality |x + 2| > 3?

Back

Split into two inequalities: x + 2 > 3 or x + 2 < -3, which simplifies to x > 1 or x < -5.

Front

What is the solution to |2x - 1| = 7?

Back

The solutions are x = -3 or x = 4.

Front

What is the solution to |x + 3| > -8?

Back

All real numbers are solutions because absolute values are always non-negative.

Front

What is the solution to |2x - 1| = -7?

Back

There is no solution because absolute values cannot be negative.

Front

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

Back

It is represented by an open interval on the number line from -a to a.

Front

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

Back

It is represented by two rays extending from -∞ to -a and from a to ∞ on the number line.