# Inequation

An inequation is a mathematical concept which represent two values

## Example of how to Solve the inequality

Solve the inequality $\mathrm{2y - 1}<5$

Solution:

$\mathrm{2y - 1}<5$

We have to treat < just like we treat = and first collect like terms

$\mathrm{2y}<5 + 1$

$\mathrm{2y}<6$

$\frac{\mathrm{2y}}{2}<\frac{6}{2}$

Answer: $y<3$

## Example of inequality

Solve the inequality $\frac{3}{2}\mathrm{n + 5}<14$

Solution:

$\frac{3}{2}\mathrm{n + 5}<14$

Collect like terms together and simplify

$\frac{3}{2}n<14 - 5$

$\frac{3}{2}n<\frac{9}{1}$

Cross muiltiply the values

$\mathrm{3n}<9 × 2$

$\mathrm{3n}<18$

$\frac{\mathrm{3n}}{3}<\frac{\mathrm{18}}{3}$

Answer: $n<6$

## Example in inequation

solve the inequation $\mathrm{1 - 2x}\le \frac{x}{2}- 9$

$\mathrm{1 - 2x}\le \frac{x}{2}- 9$

collect like terms together

$\mathrm{1 + 9}\le \frac{x}{2}+ 2x$

$\mathrm{10}\le \frac{x}{2}+ 2x$

$\mathrm{10}\le \frac{x}{2}+\frac{\mathrm{2x}}{1}$

$\mathrm{10}\le \frac{\mathrm{1\left(x\right) + 2\left(2x\right)}}{2}$

$\mathrm{10}\le \frac{\mathrm{x + 4x}}{2}$

$\mathrm{10}\le \frac{\mathrm{5x}}{2}$

$\mathrm{5x}\le \mathrm{10 × 2}$

$\mathrm{5x}\le \mathrm{20}$

$\frac{\mathrm{5x}}{5}\le \frac{\mathrm{20}}{5}$

Answer: $x\le 4$

## Solve Linear Inequalities

Given that $5x -3<7$ find x

$5x<7\mathrm{+ 3}$

$5x<10$

$\frac{x}{5}<\frac{\mathrm{10}}{5}$

$x<2$

Find the value of x from the following inequality $\frac{1}{2}\mathrm{x +}3\le 7$

$\frac{1}{2}\mathrm{x +}3\le 7$

$\frac{1}{2}x\le 7\mathrm{- 3}$

$\frac{1}{2}x\le 4$

$x\le 8$

Find the value of x given the following inequality $9 -\mathrm{3x}\ge - 3$

$9 -\mathrm{3x}\ge - 3$

$\mathrm{- 3x}\ge - 3- 9$

$\mathrm{- 3x}\ge - 12$

$\frac{\mathrm{- 3x}}{- 3}\ge \frac{\mathrm{- 12}}{- 3}$

$x\ge \frac{\mathrm{- 12}}{- 3}$

$x\le 4$

### Point to note

When the answer of an inequality is a negative, the equating symbols must be changed. E.G: if its greater than it should be changed to less than or vice versa

## How to Solve Compound Inequalities

Given the following inequality find its values $8\le \mathrm{7x - 6}\le 15$

Firstly we will derive two expression from the above inequality

$8\le \mathrm{7x - 6}\le 15$

$8\le \mathrm{7x - 6}$ and $\mathrm{7x - 6}\le 15$

$\mathrm{8 + 6}\le \mathrm{7x}$

$\mathrm{14}\le \mathrm{7x}$

$2\le x$

$\mathrm{7x - 6}\le 15$

$\mathrm{7x}\le 15 + 6$

$\mathrm{7x}\le 21$

$x\le 3$

Therefore the values are $2\le x\le 3$

Find the values of the followig Compound inequality $7\le \mathrm{4x + 3}<31$

$7\le \mathrm{4x + 3}<31$

$7\le \mathrm{4x + 3}$ and $\mathrm{4x + 3}<31$

$7\le \mathrm{4x + 3}$

$\mathrm{7 - 3}\le \mathrm{4x}$

$4\le \mathrm{4x}$

$1\le x$

$\mathrm{4x + 3}<31$

$\mathrm{4x}<31 - 3$

$\mathrm{4x}<28$

$x<7$

Therefore the values are: $1\le x<7$