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.

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