Solving Linear Inequalities

What is a linear inequality?

An inequality tells you that one expression is greater than (“>”) or less than (“<”) another
- “⩾” means “greater than or equal to”
- “⩽” means “less than or equal to”
A linear inequality only has constant terms (numbers with no letters) and terms in x (and/or a y); but no x² terms or terms with higher powers of x
- For example, 3x² > 12 is not a linear inequality (it is a quadratic inequality)
For example, 3x + 4 ⩾ 7 would be read “3x + 4 is greater than or equal to 7”.

How do I solve linear inequalities?

Solving linear inequalities is just like Solving Linear Equations
- Follow the same rules, but keep the inequality sign throughout
- If you change the inequality sign to an equals sign you are changing the meaning of the problem
When you multiply or divide both sides by a negative number, you must flip the sign of the inequality
- e.g. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)
Never multiply or divide by a variable (x) as this could be positive or negative
The safest way to rearrange is simply to add & subtract to move all the terms onto one side
You also need to know how to use Number Lines, Set Notation and deal with “Double” Inequalities

How do I represent linear inequalities on a number line?

Inequalities such as $x < a$ and $x > a$ can be represented on a normal number line using an open circle and an arrow
- For $<$ , the arrow points to the left of $a$
- For $>$ , the arrow points to the right of $a$
Inequalities such as $x \leq a$ and $x \geq a$ can be represented on a normal number line using a solid circle and an arrow
- For $\leq$ , the arrow points to the left of $a$
- For $\geq$ , the arrow points to the right of $a$
Inequalities such as $a < x < b$ and $a \leq x \leq b$ can be represented on a normal number line using two circles at $a$ and $b$ and a line between them
- For $<$ or $>$ use an open circle
- For $\leq$ or $\geq$ , use a solid circle
Disjoint inequalities such as " $x < a$ or $x > b$ " can be represented with two circles at $a$ and $b$ , an arrowed line pointing left from $a$ and an arrowed line pointing right from $b$ , and a blank space between $a$ and $b$

Solving Inequalities - Linear RN1, downloadable IGCSE & GCSE Maths revision notes

How do I represent linear inequalities using set notation?

We use curly brackets and a colon in set notation. $\{x : . . .\}$ means "x is in the set ..."

- For example; if x is greater than 3, then in set notation, $\{x : x > 3\}$

However, if x is between two values, then the two end values must be written in separate sets, using the intersection symbol, $\cap$

- For example, if x is greater than 3 and less than or equal to 5, then in set notation, $\{x : x > 3\} \cap \{x : x \leq 5\}$

Similarly, if x is less than one value or greater than another (disjoint), then the two end values must be written in separate sets using the union symbol, $\cup$

- For example, if x is less than 3 or greater than or equal to 5, then in set notation, $\{x : x < 3\} \cup \{x : x \geq 5\}$

How do I solve double inequalities?

Inequalities such as $a < 2 x < b$ can be solved by doing the same thing to all three parts of the inequality
- Use the same rules as solving linear inequalities

Exam Tip

Do not change the inequality sign to an equals when solving linear inequalities, you will lose marks in an exam for doing this.
Remember to reverse the direction of the inequality sign when multiplying or dividing by a negative number!

Worked example

(a)

Solve the inequality

- 7 \leq 3 x - 1 < 2

, illustrating your answer on a number line.

This is a double inequality, so any operation carried out to one side must be done to all three parts.
Use the expression in the middle to choose the inverse operations needed to isolate x.

Add 1 to all three parts.
Remember not to change the inequality signs.

- 6 \leq 3 x < 3

Divide all three parts by 3.
3 is positive so there is no need to flip the signs.

- 2 \leq x < 1

Illustrate the final answer on a number line, using an open circle at 1 and a closed circle at -2.

(b)

Give your answer to part (a) in set notation

Rewrite your answer using the set notation rules discussed above

{x : x \geq - 2} \cap {x : x < 1}

Worked example

Solve the inequality $5 - 2 x \leq 21$ .

Subtract 5 from both sides, keeping the inequality sign the same

$- 2 x \leq 16$

Now divide both sides by -2.
However because you are dividing by a negative number, you must flip the inequality sign

$x \geq - 8$

The final answer is normally written with the number first, but you won't be penalised for writing the x first so long as the inequality sign is the correct way around

$x \geq - 8$ or $- 8 \leq x$