## Is Number Solution to Inequality

#### Introduction

Working with inequalities gives a range of solutions for an equation. For example, if you have the inequality x > 2, the answer can be any number that is greater than 2. So, the following numbers are all solutions to the inequality x > 2: 2.01, 8, 100, 591465. 2 is not a solution, and nor is any number less than 2 (1.9, 0, -98).

#### Terms

## Lesson

To work out if a number is a solution to an inequality, you'll need to first solve the equation. You can do this either by reducing the equation so that x (or another algebraic term) is on its own, or by plugging in a value for x.

### Solving the equation

If you are given the equation:

$x + 5 < 2 + 7$

You can solve the equation until you are just left with x on one side.

First, combine the like terms on the right-hand side

$x + 5 < 2 + 7$

$x + 5 < 9$

Then subtract 5 from both sides to leave x on its own:

$x + 5 (-5) < 9 (- 5)$

$x < 4$

Therefore, x is less than 4. That means that any number less than 4 is a solution for x (3.9, 0, -986).

### Plugging in a solution

The other method you can use is to plug in a proposed number for x and see if the equation works.

If you have the equation:

$x - 3 < 7 * 2$

You can pick a number for x and see if the equation works. So, for example, we could choose 10. Plugging that in gives us:

$10 - 3 < 7 * 2$

Working that through, we get:

$7 < 14$

7 is less than 14, so 10 is a correct solution.

If we chose 100 as the number to plug in we'd get:

$100 - 3 < 7 * 2$

$97 < 14$

97 is not less than 14, so 100 is not a correct solution.

This is the simplest way to find out if a single number is a solution, but to find out a range of numbers that work, you should solve the problem to its simplest form first.

## Examples

$x + 7$ | $\leq$ | $13$ |

$(7) + 7$ | $\leq$ | $13$ |

$14$ | $\leq$ | $13$ |