Inequalities

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:

1.

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.

2.

x\geq-6

3.

x<6

4.

x\leq6

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

-7<x\leq4