Linear Inequalities (Leaving Cert Mathematics): Revision Notes

Linear Inequalities

Inequalities are mathematical statements that show a relationship of inequality between two expressions. Instead of expressing that two sides are equal (as in equations), inequalities indicate that one side is greater than or less than the other.

Solving inequalities follows the same principles as regular solving, apart from two extra rules :

• If you multiply both sides by a negative value, flip the inequality sign.
• If you divide both sides by a negative value, flip the inequality sign.

Example

infoNote

Solve the inequality

$$5x+3<16$$

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Similar to solving, we want to isolate the $x$-term :

$$\begin{align*} 5x+3&<16 & \\\\ 5x&<13 & \text{\footnotesize\textcolor{gray}{(\(-3\))}} \\\\ x&<\tfrac{13}{5} & \text{\footnotesize\textcolor{gray}{(\( \div 5 \))}} \\ \end{align*}$$

Instead of a single value for $x$, we instead solved a solution set for $x$. Only values strictly less than $\frac{13}{5}$ will satisfy the original inequality.

The values of $x$ that satisfy the inequality $5x+3<16$ (in grey)