Solving Inequalities With Variables On Both Sides

Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

Inequalities are math sentences that indicate one side of the sentence has a different value than the other (greater than or less than). Learn the process of solving these sentences, and the one twist that makes it different from solving equations.

What Are Inequalities?

Inequalities are number sentences that compare two mathematical phrases. The number sentence might tell us that the phrase on the left is greater than the one on the right, or it might indicate the phrase on the right is greater. The direction of the symbol tells which side is greater; the bigger side points toward the bigger value. Some people think of the 'V' as an alligator's mouth: the alligator will eat the most, so its mouth is open to the greater value. Here's an example:

'3x + 2 > 2x -1 ' is read: 3x plus 2 is greater than 2x minus 1.

Sometimes, the inequality indicates that one side of the sentence may be equal to the other. In this case, you will see either an extra line under the sideways 'V' or an equal sign after the inequality symbol.



is read: 3x plus 2 is greater than or equal to 2x minus 1.

Solving Inequalities

Solving inequalities is a lot like solving regular equations. The process and the rules are the same; there's just one difference, and it comes at the very end of the entire process. When solving an inequality, start the problem as if the two sides were equal and work through it just like if it were an equation.

When solving equations or inequalities, the goal is to isolate the variable on one side of the sentence and the constants on the other. Traditionally, the variable ends up on the left and the constant to the right (like: x = 3), but the reverse is fine as well.

Step one is doing the opposite of whatever you see to move the variables to one side of the symbol. This cancels out terms where they are not needed and helps to isolate the variable.

Next, do the same thing on the other side of the sign, which ensures that you don't actually change the value of the sentence. If you do the same thing on both sides of the sign, then you have not altered the total value of the sentence as a whole.

You will need to repeat these steps until you have a single variable on one side of the symbol. Remember to start with adding /subtracting and end with dividing/multiplying any values attached to the variable.

If the inequality includes an 'or equal to' symbol, make sure it also appears in your answer.

Practice Problems

Let's start with a simple equation to warm up.

x + 2 = 4

Step one: Do the opposite. Okay, so since 2 is being added to the x on the left side of the equal sign, I need to subtract 2 in order to isolate the x. That leaves me with just an x on the left.

Step two: Since I subtracted 2 on the left, I have to do the same thing on the right. 4 - 2 = 2, so the right side of the equation is now 2.

Final: x = 2

Now let's turn the equation into an inequality:

x + 2 > 4

This works exactly the same as the equation.

Step one: isolate the x by subtracting 2. Step two: subtract 2 on the other side.

Final: x > 2

Notice that the inequality points the same way as the original. This will happen all the time unless you multiply or divide by a negative number.

Inequalities With Negative Numbers

Here's another equation:

-3x = 6

Step one: The equation shows that x is being multiplied by -3, so to do the opposite we divide by -3. This isolates the x on the left by itself.

Step two: Since we divided on the left, we also divide by -3 on the right to get:


Final: x = -2

The inequality works almost the same:

-3x > 6

Step one: Do the opposite to isolate the x (so, divide by -3).

Step two: Do the same on the other side to get:


Since we divided by a negative, we switch the sign to get a final answer of x < -2.

That's the only way that inequalities differ from equations: if you must multiply or divide by a negative number to isolate the variable, then you switch the sign on the last step.

Solving Inequalities With Variables on Both Sides

Here's a trickier problem:

5x + 2 < 2x -1

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