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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.
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.
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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