What you need to know
How to Solve Inequalities
In order to solve an inequality you first need to know what the symbols mean. The following descriptions are essential in being able to solve inequalities:
- > means “greater than”,
- \geq means “greater than or equal to”,
- < means “less than”,
- \leq means “less than or equal to”.
Inequalities are not always presented to us in a straight forward way. More often than not, they’re all jumbled up – like equations often are – and therefore they need to be rearranged and solved – also like equations. Understanding how to solve equations will help with this topic, and you also need to have the basic knowledge of how to display an inequality on a number line.
Take Note:
When rearranging an inequality, you’re allowed to perform the same operation to both sides without changing it (just like an equation) with one exception: if you multiply/divide by a negative number, then the sign changes direction. In other words, < would become > and \leq would become \geq. For example, if you have
-3x \leq 6and you divide both sides by -3, then the inequality sign flips and you get
x \geq -2Example 1: Solving Inequalities
Solve the inequality 5a - 4 > 2a + 8.
So, we solve it like we would an equation, Firstly, add 4 to both sides of the inequality to get
5a > 2a + 12
Then, subtract 2a from both sides to get
3a > 12
Finally, divide both sides by 3 to get
a > 4
Note: When the question asks you to solve the inequality, your answer should be an inequality, avoid using an equals sign.
Example 2: Solving Inequalities
Solve the inequality 3x^2 - 7 <20
Firstly add 7 to both sides
3x^2 <27
The divide by 3
x^2 <9
Then finally square root
x<3
Example 3: Solving Inequalities
Solve the inequality \dfrac{4x+4}{2} > x
We need to get rid of the fraction first by multiplying by 2
{4x+4} > 2x
Then subtract 4x
4 > -2x
Then divide by -2
-2 < x
Note: Remember the sign changes direction when multiplying or dividing by a negative number
Example Questions
1) Solve the inequality 7 - 3k > -5k + 12
We solve this inequality by simply rearanging it to make k the subject,
\begin{aligned}7 - 3k &> -5k + 12 \\ 7 +2k&> 12 \\ 2k&>5 \\ k&>2.5\end{aligned}
Hence k can take any value greater than 2.5
2) Solve the inequality \dfrac{5x-1}{4} > 3x
We solve this inequality by simply rearanging it to make x the subject,
\begin{aligned}\dfrac{5x-1}{4} &> 3x \\ \\ 5x-1&> 12x \\ -1&>7x \\ x&<-\dfrac{1}{7}\end{aligned}
Hence k can take any value less than -\dfrac{1}{7}
3) Solve the inequality 2x+5 > 3x-2
We solve this inequality by simply rearranging it to make x the subject,
\begin{aligned}2x+5 &> 3x-2 \\ 7& > x \\ x&<7\end{aligned}
Hence x can take any value less than7
4)Find the range of values that satisfy the inequality, 4-3x\leq19
We solve this inequality by simply rearranging it to make x the subject in the center of the inequality,
\begin{aligned}4-3x&\leq19 \\ -3x&\leq 15 \\ 3x&\geq -15\\ x&\geq-5\end{aligned}
Hence x can take any value greater or equal to -5
5)Find the range of values that satisfy the inequality, -5<2x-3<10
We solve this inequality by simply rearranging it to make x the subject in the center of the inequality,
\begin{aligned}-5<2x&-3<10 \\ -2<2x&<13 \\ \\ -1<x&<\frac{13}{2}\end{aligned}
Hence x can take any value grteater than -1 and less than 6.5
Worksheets and Exam Questions
(NEW) Solving Inequalities Exam Style Questions
Level 4-5Inequalities - Drill Questions
Level 4-5Algebra Inequalities - Drill Questions
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