Solving inequalities is not that much different than solving equations. Instead of having an equal sign divide the two sides, there is an inequality sign.

However, there is one really important rule:

if you multiply or divide by a negative number you have to flip the inequality sign.

For example, let's look at .

	solve like you would -2x + 3 = 5
	subtract three from each side
	simplify
	divide each side by -2
	switch the sign from > to <

With this last example, if we had divided by positive two instead of negative two, we would have found that –x > 1. So, x < -1 and –x > 1 are really the same thing! That's why we have to switch the sign when we divide or multiply by a negative.

Since we divided by a -2, we switched the sign from > to <. Now, just like equations we can check our answers. Since x < -1, to check pick any number less than -1 and plug it into the original inequality (we picked -2).

Well 7 is greater than 5 so we can be pretty confident that we solved this correctly. However, unlike equations, we can't be completely sure. If you want to double check your work, that wouldn't be a horrible idea.

Look Out: only switch the inequality sign if you multiply or divide by a negative number. You do not switch it if you add or subtract a negative number.

Solving Inequalities Practice:

Example 1

Solve for y:
Here you can choose which side of the inequality to get the variables on. It's really up to you and whatever way makes more sense. In this problem, we will move the variables to the left side. add y to each side remember y is the same as 1y subtract 2 from each side simplify divide each side by 4 or notice that the sign didn't switch Since we divided each side by positive 4, we do not switch the > to <.
To check, we need to pick a number greater than 2.5 and plug it into the original inequality. Let's try 4. Well, 14 is greater than 4, so we are all good!

Example 2

Solve for x:
Just like with equations, we need to get the variable out of the denominator. We start by multiplying each side by (x-1). multiply each side by (2x - 1) the (2x - 1)'s on the left cancel distribute the -2 subtract 2 from each side simplify divide each side by -4 switch the sign from ≥ to ≤ Since we divided each side by negative 4, we switched the sign from ≥ to ≤.
To check, plug in any number greater than or equal to -½. Since it can equal -½, let's plug that in. -2 is greater than or equal to -2!

Example 3

Solve for z:
First multiply each side by -3. multiply each side by -3 switch the sign since you multiplied by -3 subtract 2 from each side simplify We switched the sign in the second step since we multiplied by a negative number.
z ≥ -20, so check by plugging in any number ≥ -20, like 0. -⅔ is smaller than 6!

Previous Example Next Example

Exercise 1

Solve for x: .

Hint

solve for x, by dividing by -1

Answer

x > -12

Exercise 2

Solve for y: .

Hint

start by multiplying each side by 2

Answer

8 ≥ y

Exercise 3

Solve for z: .

Hint

first multiply each side by (z + 3) then multiply each side by (3z)

Answer

z ≠ 1½

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