How to Solve Inequalities

By Contributor; Updated April 24, 2017
How do I solve this inequality?

Inequalities are similar to equations, you have to solve for a variable (X, Y, Z , A, B, etc...), the main difference is that with an equation you are solving for only one value (X=3, Z=4, A=-9, etc) with an inequality you are solving for a range of numbers, that means that you variable can be a number greater than, less than, greater or equal than to, less or equal than to...

For Example: If X > 3 (X is greater than 3) , X can be any value from 3.1, 3.2, 5, 7, 900, 1000 and so on.

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Inequalities Symbols

Let's remember the symbols for inequalities

Greater than >
Less than < Greater than or equal to ≥ Less than or equal to ≤

Solve the inequality like a regular equation

We have the inequality 3(X-4) ≤ X - 6. Let's solve for "X" , that means to leave "X" alone. We can solve this like a regular equation.

Solve the Parenthesis

First we need to remember PEMDAS (Please Excuse My Dear Aunt Sally). We have to solve for the parenthesis. Let's multiply 3 times X, and 3 times -4

Move the variables to the left side

Once we do the parenthesis, 3x - 12 ≤ X -6, let's move the "X" from the right to the left side, we do this by adding "X" to both sides.

Move the numbers to the right side

Our inequality looks like this 2X - 12 ≤ X -6. Now we have to move -12 from the left to the right side, let's add 12 to both sides.

Divide by 2 both sides

Our main goal is to leave "X" alone, 2 is multiplying X, let's eliminate him from the left side by dividing both sides by 2

Final Answer

Our result is X ≤ 3, that means that the value of X has to be a number less than or equal to the number 3. For example 3, 2, 1, 0 -1, -2, -3 and so on. We can also write our answer like this (-∞, 3], we use always parenthesis for the infinitive symbol , and we use a bracket because our inequality is less than or equal to. If our equation was 3(X-4) < X -6, then our answer would be ( -∞, 3) with a parenthesis, this means that X can not be 3, it has to be less than 3, for example 2.99, 2.50, 0, -1, -2, -3 . Conclusion. If you have an inequality with the equal symbol (≤≥) , you have to use the bracket [], if you have the inequality without the equal symbol (<>), then you have to use parenthesis ()

Tip

Check our other articles about how to solve equations if you are having trouble solving for "X" Understand the inequality symbols If you divide by a negative number both sides, your inequality symbol will flip to the opposite side. For Example: -3X>6, -3X/-3 > 6/ -3, then X<-2, if you have doubts, just plug your answer and make sure makes sense, in our example X has to be less than -2 , so -3(-3) > 6, 9> 6, if you do not flip the inequality then your answer would be wrong.

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