What Is An Inequality?

When you start learning algebra, an equals sign is used to signify, quite literally, the two things are equal to each other. For example 3 = 3, 5 = 3 + 2, apple = apple, pear = pear and so on, which are all examples of equations. By comparison, an inequality gives you two pieces of information: First, that the things being compared are ​not​ equal, or at least not always equal; and second, in what way they are unequal.

An inequality is written exactly as you'd write an equation, except that instead of using an equals sign, you use one of the inequality signs. They are ">" a.k.a. "greater than," "<" a.k.a. "less than," "≥" a.k.a. "greater than or equal to" and "≤" a.k.a. "less than or equal to." Technically the first two symbols, > and <, are known as strict inequalities because they don't include any option for the two sides of the inequality to be equal. The signs ≥ and ≤ denote the possibility that the two sides are equal ​and​ unequal.

A visual representation – that is, a graph – of an inequality is another way of visualizing what an inequality really means. Graphing inequalities is also something you'll be asked to do in math class. Imagine the following equation:

\(x = y\)

If you were to graph this out, it'd be a diagonal line passing straight through the origin, angled up and right with slope of 1 or, if you prefer, 1/1. All the possible solutions for the equation lie on that line, and only on that line.

But what if instead of an equation, you had the inequality

\(x ≤ y\)

This particular inequality symbol would be read as "less than or equal to" and tells you that ​x​ = ​y​ is a possible solution, along with every combination where ​x​ is less than ​y​.

So the line representing ​x​ = ​y​ remains a possible solution, and you'd draw it in as usual. But you'd also shade in the area to the left of the line, because any value where ​x​ is less than ​y​ is also included in your solutions.

If instead of ​x​ ≤ ​y​ you had the strict inequality ​x​ < ​y​, you would graph it exactly the same as ​x​ ≤ ​y,​ except that because ​x​ = ​y​ is no longer an option, you wouldn't draw that line in solidly. Instead, you'd draw ​x​ = ​y​ in as a dashed or broken line, showing that although it's not part of the solution set, it is still the border between the valid solution set (in this case, to the left of your line) and the non-solutions on the other side of the line.

For the most part, solving inequalities works exactly the same as solving equations. For example, if you were faced with the simple equation

\(2x = 6\)

you'd divide both sides by 2 to arrive at the answer ​x​ = 3.

You'd do the same if you were, instead, faced with the same numbers as an inequality: Say, 2​x​ ≥ 6. You'd divide both sides by 2 and arrive at the solution ​x​ ≥ 3 or, to write it out in plain English, ​x​ represents all numbers greater than or equal to 3.

You can also add and subtract numbers on both sides of an inequality, just as you do with equations, or divide by the same number on both sides.

But there is one notable exception to watch out for: If you multiply or divide both sides of an inequality by a negative number, then you have to flip the direction of the inequality sign. For example, consider the inequality -4​y​ > 24.

To isolate ​y​, you'll need to divide both sides by -4. That's your trigger to switch the direction of the inequality sign. So after dividing, you have:

\(y < -6\)

Note that the set of solutions for the inequality just given include −7, −8, −7.5, −9.23 and an infinite number of other solutions that are less than −6, but not −6 itself, because the inequality sign doesn't have the extra bar for "or equal to." So to check your work, make sure you substitute values from your solution set.

If you substitute −6 into the original inequality you'd end up with −4 × −6 > 24 or 24 > 24, which makes no sense. Nor should it, since −6 is not included in the solution set. But if you were to start substituting values that ​are​ included in the solution set, such as −7, you'd get valid results. For example:

\(-4 × -7 > 24\)

which simplifies to:

\(28 > 24\)

which is a valid result.