What is an inequality?

This is an example of an inequality.

2x + 4 < 92

It is a statement that a certain expression or number is not equal to or may be equal to another expression or number. The < sign in the inequality states that 2 times a number (x) plus 4 is less than 92. The smaller end of the sign represents the smaller item, and the larger end represents the larger end, of course. For the purposes of naming, they are typically read from left to right. (< being the “less than” symbol and > being the “greater than” symbol, though they mean almost the same thing depending on the positioning of numbers.)

This is a compound inequality. It uses more than one sign.

4<x<9

This indicates that x is greater than 4, but smaller than 9.

There is also and , which represent “greater than or equal to” and “less than or equal to” symbols. They are used like so.

6x – 9 42 + x

This states that 6 times a number minus 9 is greater than or equal to 42 plus a number.

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How to solve an inequality

Solving an inequality is similar to solving an equation, which itself is similar to solving polynomials. Let’s take my first example inequality from before.

2x + 4 < 92

From here, one can perform legal moves to both sides if the inequality. If you did not read my equation solving post, a legal move is an operation applied to both sides of the equation/inequality. In this case, we’ll subtract 4.

2z + 4 – 4 < 92 – 4

2x < 88

From here, you divide the two numbers by 2 in order to isolate the variable (x). This is so that we can determine what number that the variable has to be bigger than or smaller than.

2x 2 = 1x (x)

88 2 = 44

x < 44

This means that x is greater than 44.

Inequalities can also include fractions, decimals, coefficients and negative numbers. Here’s an inequality with all three.

(5x+10)>2.75-4x

First, I’ll distribute the coefficient to simplify the inequality. That would be multiplying 5x and 10 by

2.5x+5>2.75-4x

Now, we’ll convert the decimals to fractions for ease of handling. The will be doubled to give it a common denominator with

x+5>-4x

I’ll add 4x to nullify the -4x on the right side of the inequality, and subtract 5 to nullify the 5 on the left side.

–x>

Now, we divide both sides of the inequality by –. Because it is negative, the > sign must be inverted into < and the will become negative. – will become positive x.

x<-

We can see that x is smaller than –

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How to check a solution

Checking the solution is checking if the variable can in fact be less than, greater than or equal to the supposed solution. Let’s take my example of 2x + 4 < 92 again. We found that x is smaller than 44, so let’s try making x 43.

2(43) + 4 < 92

We multiply 43 by 2.

86 + 4 < 92

86 + 4 = 90

90 < 92

This confirms that the solution is correct. Let’s go back to my other example of (5x+10)>2.75-4x now. The supposed solution was –. For the ease of verification, I’ll convert fractions to decimals.

0.5(5(-1.5)+10)>2.75-4(-1.5)

0.5(-7.5+10)>2.75-6

-3.75+10>2.75-6

6.25>-3.25

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Graphing inequalities

Inequalities can be graphed on a number line. Here is 6<x on a graph.

The circle is called the boundary point. It separates possible solutions from the other numbers. The fact that the circle is open (an open dot) means that 6 is not a possible solution. If the inequality were 6x however…

The coloured circle (closed dot) indicates that 6 is a possible solution.

If the number line were to be describing a compound fraction, for example, -2<x11, it would look like this.

As you can see, the circle over the -2 is open, but not the one over the 11.

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