How To Solve Inequalities

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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Step 1

Let's remember the symbols for inequalities

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

Step 2

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.

Step 3

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

Step 4

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.

Step 5

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.

Step 6

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

Step 7

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 ()