# Inequality with no solutions

An inequality with no solutions is easy to solve or easy to recognize. See a few examples below.

Example #1

Solve x - x > 6

x - x > 6

x - x = 0, so we get 0 > 6

Since 0 is never bigger than 6, this inequality has no solutions.

Example #2

Solve 2x + x - 5 > 3x + 4

2x + x - 5 > 3x + 4

Add 5 to each side of the inequality

2x + x - 5 + 5 > 3x + 4 + 5

2x + x > 3x + 9

Simplify by adding x and 2x

3x > 3x + 9

Subtract 3x from each side of the inequality

3x - 3x > 3x - 3x + 9

0 > 9

Since 0 is never bigger than 9, this inequality has no solutions.

## Compound inequality with no solutions

Example #3

Solve x < 2 and x > 9

Since x cannot be smaller than 2 and bigger than 9 at the same time, this compound inequality states something that is false or something that can never happen.

Therefore, this compound inequality has no solutions.

### Absolute value inequality with no solutions

Example #4

Solve |x| < -3

The absolute value of a number is always positive.

For example, suppose x is positive and choose x = 1

|1| = 1 and 1 is not smaller than -3

Suppose x is negative and choose x = -1

|-1| = 1 and 1 is not smaller than -3

Therefore, |x| < -3 has no solutions

No matter what number you choose for x, when you take the absolute value, it will never be smaller than -3.

In general, if |x| < a and a is a negative number, |x| < a has no solutions.

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

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