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Equations and inequalities

Linear inequalities

Linear inequalities

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Summary

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 xxx​ 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 xxx for which:

i)  6x−3<2x+76x-3 \lt 2x+76x−3<2x+7

ii) 5x+6≤12−x5x+6 \le 12-x5x+6≤12−x


i) Rearrange to get xxx on one side:

4x<104x \lt 104x<10​​


Solve for xxx:

x<2.5‾\underline{x \lt 2.5}x<2.5​​​


ii) Rearrange and solve for xxx:

6x≤66x \le 66x≤6


x≤1‾\underline{x \le 1}x≤1​​​​


Note: The solutions for a linear inequality can also be written in set notation. x≤1x \le 1x≤1 can be written as {x:x≤1}\{x:x \le 1\}{x:x≤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>−4x \gt -4x>−4 and x≤3x \le3x≤3, the solutions which satisfy both of these simultaneously are −4<x≤3-4 \lt x \le 3−4<x≤3. This can also be written as {x:−4<x≤3}\{x:-4 \lt x \le 3\}{x:−4<x≤3} or {x:x>−4} ∩ {x:x≤3}\{x:x \gt -4\}\ \cap\ \{x:x\le3\}{x:x>−4} ∩ {x:x≤3}.


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


Example 2

Find the set of values of xxx for which:

i) 3x−8>x+53x - 8 \gt x+5 3x−8>x+5 and 2x+6>x+82x+6 \gt x+82x+6>x+8​

ii) 2x−5>92x-5 \gt 92x−5>9 and 4x+5<10+x4x +5 \lt 10+x4x+5<10+x​


i) Solve the first inequality:

3x−8>x+52x>13x>6.5\begin{aligned}3x - 8 & \gt x+5\\2x & \gt13\\ x & \gt 6.5\end{aligned}3x−82xx​>x+5>13>6.5​

​​​​

Solve the second inequality:

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


The solutions for both inequalities overlap when xxx is greater than 6.56.56.5:


Therefore, {x:x>6.5}‾\underline{\{x:x\gt6.5\}}{x:x>6.5}​


ii) Solve the first inequality:

2x−5>92x>14x>7\begin{aligned} 2x-5 & \gt9\\ 2x & \gt14\\ x & \gt7 \end{aligned}2x−52xx​>9>14>7​


Solve the second inequality:

4x+5<10+x3x<5\begin{aligned}4x+5&\lt10+x\\3x&\lt5\end{aligned}4x+53x​<10+x<5​


x<53\begin{aligned}x & \lt \dfrac53\end{aligned}x​<35​​​​​​


The solutions for the inequalities do not overlap:


Therefore, x<53 or x>7‾\underline{x\lt \dfrac53\ or\ x\gt7}x<35​ or x>7​​



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FAQs - Frequently Asked Questions

How do you solve linear inequalities?

Linear inequalities can be solved using similar methods to linear equations.

Do the solutions for linear inequalities have to be written together?

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

Are there any other ways to write solutions for inequalities?

The solutions for a linear inequality can also be written in set notation.

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