## Linear inequalities

Inequalities are mathematical statements similar to equations, but with the equals sign replaced with one of the following four inequality symbols

Symbol | Meaning |

$\lt$ | less than: $a\lt b$ - the value of $a$ is strictly less than the value of $b$ |

$\gt$ | greater than: $a\gt b$ - the value of $a$ is strictly greater than the value of $b$ |

$\leq$ | less than or equal to : $a\leq b$ - the value of $a$ is either less than the value of $b$ or possibly equal to the value of $b$ |

$\geq$ | greater than or equal to : $a\geq b$ - the value of $a$ is either greater than the value of $b$ or possibly equal to the value of $b$ |

The method of solving an inequality is effectively identical to the process of solving an equation; but with a couple of caveats

- multiplication or division by a negative number changes the direction of an inequality ($\gt$ becomes $\lt$ and vice versa)
- The solution is still an inequality

## Linear inequalities 1

Solve the following inequality:

$$ 3x - 10 \lt 23 $$## Linear inequalities 2

Note that the coefficient of the variable is negative so some care needs to be taken

**Approach 1**

Avoiding multiplication and division by negative numbers:

$$\begin{align}4 - 2t &\leq 20\\ 4 &\leq 20 + 2t \\ -16 &\leq 2t \\ -8 &\leq t\end{align}$$

**Approach 2 **

Multiplication or division by a negative number changes the direction of the sign

$$\begin{align}4-2t &\leq 20 \\ -2t &\leq 16 \\ t &\geq -8 \end{align}$$

Justification of the change of orientation of the sign requires a little thought, but we see both methods return an equivalent result.

## Linear inequalities 3

Solve the following system of inequalities:

$$ \begin{align} 4x + 6 &\gt 22 \\ 40-2x &\geq 10 \end{align} $$In addition, state the cardinality (number of members) of the set of integers that satisfy the inequalities.

The first inequality reduces to $$ \begin{align} 4x + 6&\gt 22 \\ 4x &\gt 16 \\ x &\gt 4 \end{align} $$

The second yields $$ \begin{align} 40 - 2x &\geq 10\\ -2x &\geq -30 \\x &\leq 15 \end{align} $$

Hence the solution of the set of inequalities is $$4\lt x\leq 15 $$

The set of integer solutions is $\left\{5,6,7,8,9,10,11,12,13,14,15\right\}$ which has a cardinality (number of elements) of 11