Inequalities

GCSELevel 6-7Cambridge iGCSE

Inequalities Revision

Inequalities

When we deal with equations, the equals sign tells us that each side of the equation are equal to each other. Another type of equation is an inequality, where the sides of the equation do not equal each other. The inequality sign may tell us one side of the equation is smaller or larger than the other, or smaller than or equal to, or larger than or equal to, each other. You will need to understand how to derive and solve inequalities.

We can also display inequalities on a number line. You will need to understand how to use and interpret inequalities on a number line.

Inequality Symbols

There are $4$ symbols used in inequalities, in place of an equals sign in an equation:

$>$ greater than

Example: $4 > 3$ means $4$ is greater than $3$

$<$ less than

Example: $3 < 4$ means $3$ is less than $4$

$\geq$ greater than or equal to

Example: $4\geq4$ means $4$ is greater than or equal to $4$ ( $6\geq4$ is also an example that works)

$\leq$ less than or equal to

Example: $2\leq3$ means $2$ is greater than or equal to $2$ ( $3\leq3$ is also an example that works)

We often see these sign used in algebraic inequalities:

$3x < 4$ means $3x$ is less than $4$

Solving Inequalities – Basics

For the most part, we can treat and solve inequalities just as we do with equations (we will talk about the exception later).

Example: solve $3x+3\leq12$ ,

We would treat this as an equation and start by minusing $3$ from both sides:

$3x\leq9$

And now, divide both sides by $3$ :

$x\leq3$

Meaning $x$ is less than or equal to $3$ . This inequality is now solved.

Solving Inequalities with 2 Signs

You may also come across inequalities with $2$ signs.

Example:

$12<2x+6<24$

This means, $2x+6$ is between $12$ and $24$ . Or $2x+6$ is greater than $12$ but less than $24$ .

We can just as easily solve inequalities like this, in the same way as solving equations, but whatever is done to one side, must be done to all $3$ sides:

$12<2x+6<24$

Start by subtracting $6$ from each side to get $2x$ on its own:

$6<2x<18$

And then dividing by $2$ to get $x$ on its own:

$3<x<9$

Hence, $x$ is greater than $3$ but less than $9$

Note: the two signs may not always be the same (for example $12>x\geq2$ , meaning $x$ is less than $12$ but greater than or equal to $2$ ) but these are solved using the same method.

Solving Inequalities: the Exception

There is one case where solving inequalities differs from solving equations, and this occurs when multiplying or dividing by a negative number.

RULE: if you multiply or divide by a negative number, the inequality sign changes direction

When we flip a sign (this means the same as the sign changing direction):

$< \rightarrow >\\$

$> \rightarrow <\\$

$\leq \rightarrow \geq\\$

$\geq \rightarrow \leq\\$

Example: solve $19<-8x-5<27$

Start as usual by adding $5$ to each side:

$24<-8x<32$

Now, we want to isolate the $x$ , so we need to divide by $-8$ but as we are diving by a negative number, we need to flip the signs to $>$ ,

$-3 > x > -4$

Meaning $x$ is less than $-3$ but greater than $-4$

Inequalities on a Number Line

We can display inequalities on a number line.

Note: a filled circle, $\bullet$ is used for the inequalities less than or equal to, and greater than or equal to. Whereas an open circle, $\circ$ , is used for less than or greater than (without the equal to).

Examples:

$x>-6$

You can see the arrow is pointed towards values greater than $-6$ , because the inequality sign is a greater than, and the circle is open because it is not greater than or equal to.

$x\geq-6$

$x<6$

$x\leq6$

Number lines can also be used for inequalities with $2$ signs, for example:

$-7<x\leq4$

Example 1: Rearranging and Solving Inequalities

Solve the following inequality,

$42+x\leq\dfrac{1}{4}x\leq51+x$

[3 marks]

Let’s subtract $x$ from each side to get rid of the $x$ ‘s on the outside of the inequality:

$42\leq-\dfrac{3}{4}x\leq51$

Now, let’s divide by $-\dfrac{3}{4}$ to get $x$ on its own. Remember – when dividing by a negative number we need to use the opposite signs:

$-56\geq x\geq-68$

Example 2: Number Lines

Express the following inequality on a number line,

$-3\leq x<5$

[2 marks]

So, as the first sign means less than or equal to, the circle on the number line will be filled, but the second sign, less than, will be an empty circle: