Absolute Value Inequalities

Front

What is an absolute value inequality?

Back

An absolute value inequality is an inequality that contains an absolute value expression, which measures the distance of a number from zero on the number line.

Front

How do you solve the inequality |x + 3| > 8?

Back

Split into two cases: x + 3 > 8 or x + 3 < -8. This leads to x > 5 or x < -11.

Front

What does the solution x > 5 or x < -11 represent?

Back

It represents all values of x that are either greater than 5 or less than -11.

Front

What is the solution to the inequality |x + 4| - 8 > 2?

Back

First, rewrite as |x + 4| > 10. Then split into two cases: x + 4 > 10 or x + 4 < -10, leading to x > 6 or x < -14.

Front

What does the solution x > 6 or x < -14 indicate?

Back

It indicates that x can take any value greater than 6 or any value less than -14.

Front

What is the result of solving |2x - 1| = -7?

Back

There is no solution because absolute values cannot be negative.

Front

What does the inequality -5|x - 4| < 20 imply?

Back

Divide by -5 (reversing the inequality): |x - 4| > -4. Since absolute values are always non-negative, this is true for all x.

Front

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

Back

This inequality is always true for all real numbers since absolute values are always greater than or equal to zero.

Front

What is the general approach to solving absolute value inequalities?

Back

1. Isolate the absolute value expression. 2. Set up two separate inequalities based on the definition of absolute value. 3. Solve each inequality.

Front

What is the difference between |x| < a and |x| > a?

Back

|x| < a means x is within a distance a from 0, while |x| > a means x is outside that distance.