Solving Inequalities:

Example1: Solve for x

(x – 2) / -3 < x + 4

Answer:

In this case, notice that the (x – 2) is being divided by -3. A good first step is to multiply all terms by -3 to remove this fraction.

NOTE: When multiplying or dividing by a negative value across and inequality, you must switch the direction of the inequality.

-3* (x – 2) / -3 > -3*x +  -3*4

 

x – 2 > -3x – 12

 

Notice that when we multiply all terms by -3, it removes (reduces always) the -3 on the denominator on the left. It also switches the direction of the inequality. The “math” reason that the inequality swaps directions is because we are multiplying (in this case) by a negative value which causes the inequality to be the “opposite”.

 

Now, add 2 to both sides

x – 2 + 2 > -3x – 12 + 2

x > -3x – 10

 

Next, add 3x to both sides.

x + 3x > -3x – 10 +3x

4x > -10

 

Finally, divide both sides by 4

x > -10/4

x > -2.5

 

Example 2: Solve for x

3 – (x +4) < -2x + 7

Answer:

First, simplify both sides. Be sure to distribute the negative times the (x + 4) to get –x – 4

3 – x – 4 < -2x + 7

Combine like terms

 

-1 – x < -2x + 7

Next, add 2x to both sides

-1 – x + 2x < -2x + 7 + 2x

-1 + x < 7

 

Add 1 to both sides

-1 + x + 1 < 7 + 1

x < 8

 

Notice that in this example, because we do not multiply or divide by a negative value, we do not switch the direction of the inequality.