## Solving Inequalities

#### Introduction

Inequalities are expressions that show that two numbers are not equal to one another, but one is larger than another. For example, if we have two brothers, John and Robert, and John is older than Robert, we can express that as an inequality:

John's age > Robert's age

The '>' tells us that the value on the left is larger than the value on the right. We can also write it as:

Robert's age < John's age.

The '<' tells us that the value on the left is smaller than the value on the right.

To remember which one is which, remember that the 'big' end of the symbol always points to the bigger value, and the 'small' end always points at the smaller value.

#### Terms

## Lesson

To solve inequalities, we treat them the same as equations linked by an equals sign. We need to get the x isolated on its own. Using the following equation:

$x - 3 < 2 + 5$

Firstly, we combine the like terms on the same side of the equation:

$x - 3 < 2 + 5$

$x - 3 < 7$

Then we get rid of the -3 by adding 3 to both sides:

$x -3 (+3) < 7 (+3)$

$x < 10$

Therefore we have solved the inequality. x is less than 10, so any number less than 10 would solve the equation (e.g. 9, 3.5, -1,000).

If we end up with negative numbers on both sides of our equality, we need to reverse the sign to solve.

For example:

$5-x > 3 - 2 + 1$

We first combine like terms on the same side of the equation:

$5 + 3x > 2x + 3 - 2 + 1$

$5 + 3x > 2x$

Then we subtract -3x from both sides to get the like terms on the same side

$5 + 3x (-3x) > 2x (-3x)$

$5 > -x$

We don't want x to be a negative, so we multiply everything in the equation by -1. This makes everything positive into a negative and everything negative into a positive. However, when we do this, we also need to reverse the direction of the inequality:

$5 > - x$

$-5 < x$

x is, therefore, greater than -5, so any number larger than -5 will solve the equation. (e.g. -4, 0, 897)

≥ and ≤ work in exactly the same way as > and < except the answers can be the number on the other side of the equation.

For example:

x ≤ 8

x could be any number larger than 8 (9, 28, 8432) **AND** could be 8.

Think about it like this: ≥ is > as well as = ≤ is < as well as =

## Examples

$x + 4$ | $\leq$ | $6$ |

$-4$ | $=$ | $-4$ |

$x$ | $\leq$ | $2$ |

$\dfrac{-x}{2} + 6$ | $\leq$ | $-3$ |

$-6$ | $=$ | $-6$ |

$\dfrac{-x}{2}$ | $\leq$ | $-9$ |

$*-2$ | $=$ | $*-2$ |

$x$ | $\leq$ | $18$ |