What is an inequality?

An inequality is almost the same as an equation, but in an inequality we don't have an equals sign, $=$. We have a greater-than or less-than sign, $<$ or $>$. Actually exactly the same rules apply as for equations. We solve inequalities the same way we solve equations.

If that sounds like a relief, it should. Almost everything you know about equations carries straight over. There's one new thing to remember, and we get to it below.

The four signs

First we have to look at the signs that take the place of the equals sign:

$$x > \; : \; x \text{ is greater than}$$

$$x \geq \; : \; x \text{ is greater than or equal to}$$

$$x < \; : \; x \text{ is less than}$$

$$x \leq \; : \; x \text{ is less than or equal to}$$

These are the kinds of sign we can have in an inequality. If it said, for example,

$$x \geq 5$$

that would mean x is greater than or equal to 5. So x can be anything from 5 up to something infinitely large. That's the first thing that's different from an equation: the answer isn't one number, it's a whole range of numbers.

Solving an inequality

When we solve inequalities we do it, as said, just like we solve equations, but we have to watch out for one small difference. When we multiply or divide by a negative number on both sides, the sign turns round. If it was a greater-than, it becomes a less-than, and the other way round.

As long as we don't do that, nothing changes. Take the inequality

$$5x + 6 > 2x + 7$$

We subtract $2x$ on both sides, subtract 6 on both sides, and divide by 3 on both sides, exactly as we would with an equation, and we end up with

$$x > \frac{1}{3}$$

That x has to be greater than $\tfrac{1}{3}$ just means that every value from $\tfrac{1}{3}$, where $\tfrac{1}{3}$ itself isn't included, up to an infinitely large number is a solution to this inequality. If $\tfrac{1}{3}$ was allowed to be a solution too, it would have said $\geq$ instead of $>$.

So, as we see, it's exactly like solving equations. But we just have to remember the rule that if we multiply or divide by a negative number, we have to flip the inequality sign. Every step is in the guide how to solve linear inequalities.

Double inequalities

Double inequalities are when we have two inequality signs in one inequality. It could look like this:

$$3x + 7 > x + 3 > 4x - 2$$

When we solve one of these, we actually pretend we have 2 inequalities, and solve them one at a time. Whatever stands between the two signs goes into both. The walkthrough is in double inequalities, together with the way we use a double inequality like $2 < x < 7$ to mark out a piece of a graph.

Common misunderstandings

• "Solving an inequality is a whole new thing to learn." It isn't. An inequality is almost the same as an equation, and the rules are the same. There's exactly one new rule.
• "The answer is a single number." No. $x \geq 5$ means x can be anything from 5 up to something infinitely large. The answer is a range.
• "$>$ and $\geq$ mean the same." They don't. With $>$ the number itself is left out, with $\geq$ it's included. $x > \tfrac{1}{3}$ doesn't include $\tfrac{1}{3}$.
• "Multiplying both sides by a negative number leaves the sign alone." It doesn't. That's the one rule: multiply or divide by a negative, and the sign turns round.
• "A double inequality is a special new kind of problem." It's just two inequalities stapled together. Split it in two and solve each one.

Related guides

• How to solve linear inequalities
• Double inequalities
• The machinery it all borrows from: What is an equation?