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Solving Inequalities Using Addition, Subtraction, Multiplication & Division

Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Solving inequalities is almost identical to solving equations. By using the inverse operation for the inequality, variables are quickly isolated to find solutions.

What is an Inequality?

I'm sure you've heard of an equation before. It's a mathematical sentence showing equal amounts on either side of an equals sign. It is like a see-saw that is stuck perfectly balanced. The equals sign (=) is at the center of the see-saw.

pictorial example of an equation

Knowing that the prefix in means not helps us to understand that an inequality means that the two sides of a mathematical sentence are not equal. In this case, the see-saw is no longer stuck in the middle, one side is higher than the other; one side of the inequality has a higher value than the other. In other words, they are not equal.

pictorial example of an inequality

Reading Inequalities

Before you can read an inequality, you have to 'understand' the language. In this case, you must know how to read the four inequality symbols:

Did you notice the words 'or equal to'? This part of the inequality symbol comes from the line under the greater than and less than symbol.

You read an inequality from left to right, just like equations. The relationship between the sides of the sentence depends on the direction of the inequality symbol. One simple way to remember the relationship is to think of the inequality symbol as an alligator; it always has its mouth open toward the larger value.

inequality

This inequality is read: 3x + 2 is greater than or equal to 2x - 1. Because this sentence has a 'greater than or equal to' symbol, it is both an inequality and an equation (it is solved when the value of x is either equal to or greater than the value on the other side of the sign).

Solving Inequalities

Solving inequalities works exactly like solving equations except that the final answer will include a range of values instead of just one value.

Before we begin with the actual steps involved in solving an inequality, we must learn the most important rule in solving an inequality: The problem must always be balanced - the absolute value of the inequality must be maintained. What this means is that whatever action you perform on one side of the inequality symbol, you must also perform on the other side.

Now that you know that all-important rule, let's continue. In order to solve an inequality, you must isolate the variable, a symbol (usually a letter) that is representing a currently unknown number. You isolate the variable by using addition, subtraction, multiplication, or division to 'cancel out' the numbers on the variable side until the variable stands alone on one side of the inequality and all the numbers have been combined on the other side.

Whew... That is a mouthful. Let's break it down into two very easy steps:

Step 1: Do the inverse, opposite, of whatever mathematical operation is being performed on the variable side of the problem.

Step 2: Do the same step on the other side of the inequality (remember, we must keep the problem balanced).

Repeat these two steps until the variable is all alone on one side of the inequality symbol.

Let's consider this inequality:

3x + 1 > 42

We see that on the left side (which contains the variable x), 1 is being added to 3x. To isolate the variable, we must first get rid of that 1. The rule is to do the opposite, right? So, what is opposite of adding? Subtracting! Our second rule is to do the same thing on both sides, so we'll subtract 1 from both sides.

First: Subtract 1 from both sides of the symbol. We now have: 3x > 41

This helped, but the variable is still not alone, but rather it is being multiplied by 3. The opposite of multiplication is division, so we divide both sides by 3 in order to get the variable isolated.

Second: Divide both sides by 3 to get: x > 13.67

Check: the variable x is isolated; the problem is solved. The solution is anything larger than 13.67.

Did you notice that we started with the addition/subtraction and then moved to multiplication/division? We isolated the variable term (number/variable combinations separated by operations such as +, -, x, and ÷) first by dealing with the addition/subtraction portion of the process. Then we moved to the multiplication/division portion to completely isolate the variable itself.

A More Complex Example

What if you have a variable on both sides of the symbol?

4x - 1 < 2x + 3

Here, there is a variable on both sides of the symbol. Don't worry; we still follow the same rule with the same goal - to isolate the variable.

First: Perform the opposite operation to isolate the variable term on the left.

In this case, we will first subtract 2x from both sides which leaves us with 2x - 1 < 3

We will then add 1 to both sides. The problem now reads 2x < 4

Second: Perform the opposite operation to get the variable itself isolated

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