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Question:

Solve for x:
4x – 12 ≥ 6x + 2

A x ≤ -7
Explaination


When solving inequalities remember to use:
Inverse Operations – operations that undo one another. So if subtraction is present, we use addition, etc.
What we do to one side we MUST do to the other
Our goal is to ISOLATE the variable which means to have JUST one of the variables.
Step 1: Get all x-terms to one side of the inequality
In this inequality, we notice that there are x-terms on both sides of the inequality symbol so we can start by applying the inverse of one of the x-terms on both sides of the inequality.
We see that there is a 6x on the right side of the inequality, so we can apply the additive inverse of 6x on both sides so that we only have x-terms on the left side of the equation. Therefore we are going to subtract 6x from both sides.
4x – 12 – 6x ≥ 6x + 2 – 6x
Which then becomes:
-2x – 12 ≥ 2
Step 2: Solve the Inequality
When we solve inequalities, we can start by applying the inverse of the constant term on both sides of the equation. The constant is the term with no variable attached to it.
We see that 12 is being subtracted from the -2x, the -12 is the constant term because there is no x attached, so we have to undo the subtraction by using addition. Therefore we are going to add 12 to both sides.
-2x – 12 + 12 ≥ 2 + 12
Which then becomes:
-2x ≥ 14
Now we can solve:
Since -2 is being multiplied by x, we have to undo the multiplication by using division. Therefore we are going to divide both sides by -2.
(-2x/-2) ≥ (14/-2)
There is an important rule to remember when we divide inequalities by a negative number.
When an inequality is divided by a negative number the inequality symbol switches direction as shown below:
1x ≤ -7
Since 1x and x are the same thing, our final answer is:
x ≤ -7