Linear inequalities

In a nutshell

Linear inequalities can be solved using similar methods to linear equations. Sometimes you will need to find the set of values for which two inequalities are true.

Linear inequalities with one solution

When an inequality has one solution, you rearrange the inequality to make $$x$$​ the subject. The solution of an inequality is the set of all real numbers which make the inequality true.

Example 1

Find the set of values of $$x$$ for which:

i)  $$6x-3 \lt 2x+7$$

ii) $$5x+6 \le 12-x$$

i) Rearrange to get $$x$$ on one side:

$$4x \lt 10$$​​

Solve for $$x$$:

$$\underline{x \lt 2.5}$$​​

ii) Rearrange and solve for $$x$$:

$$6x \le 6$$

$$\underline{x \le 1}$$​​​

Note: The solutions for a linear inequality can also be written in set notation. $$x \le 1$$ can be written as $$\{x:x \le 1\}$$.

Linear inequalities with two solutions

When there are two solutions to a linear inequality, the set of values can be written together or separately. 

When a solution set is $$x \gt -4$$ and $$x \le3$$, the solutions which satisfy both of these simultaneously are $$-4 \lt x \le 3$$. This can also be written as $$\{x:-4 \lt x \le 3\}$$ or $$\{x:x \gt -4\}\ \cap\ \{x:x\le3\}$$.

When a solution set is $$x \lt 1$$ or $$x \ge 6$$, there are no overlaps with the solutions , therefore these are written seperately as $$x \lt 1$$ or $$x \ge 6$$. This can also be written as $$\{x:x \lt 1\}\ \cup\ \{x:x \ge 6\}$$.​

Example 2

Find the set of values of $$x$$ for which:

i) $$3x - 8 \gt x+5$$ and $$2x+6 \gt x+8$$​

ii) $$2x-5 \gt 9$$ and $$4x +5 \lt 10+x$$​

i) Solve the first inequality:

$$\begin{aligned}3x - 8 & \gt x+5\\2x & \gt13\\ x & \gt 6.5\end{aligned}$$

​​​​

Solve the second inequality:

$$\begin{aligned} 2x+6 & \gt x+8\\ x & \gt 2 \end{aligned}$$

The solutions for both inequalities overlap when $$x$$ is greater than $$6.5$$:

Therefore, $$\underline{\{x:x\gt6.5\}}$$

ii) Solve the first inequality:

$$\begin{aligned} 2x-5 & \gt9\\ 2x & \gt14\\ x & \gt7 \end{aligned}$$

Solve the second inequality:

$$\begin{aligned}4x+5&\lt10+x\\3x&\lt5\end{aligned}$$

$$\begin{aligned}x & \lt \dfrac53\end{aligned}$$​​​​

The solutions for the inequalities do not overlap:

Therefore, $$\underline{x\lt \dfrac53\ or\ x\gt7}$$​