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Basic Algebra

Basic Algebra

At a Glance - Solving Inequalities

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 -2x + 3 > 5.

-2x + 3 > 5solve like you would -2x + 3 = 5
-2x + 3 - 3 > 5 - 3subtract three from each side
-2x > 2simplify
-2x/-2 > 2/-2divide each side by -2
x < -1switch 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).

-2(-2) + 3 > 5

4 + 3 > 5

7 > 5

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.

Did you forget what inequalities were and became confused by this section? Well, here's a helpful refresher!

Example 1

Solve for y:

3y + 2 > 12 - y

Example 2

Solve for x:

4/(2x-1) ≥ -2

Example 3

Solve for z:

z + 2 (greater than or equal to) -18

Exercise 1

Solve for x: -x/4 < 3.

Exercise 2

Solve for y: 15 ≥ (5y-10)/2.

Exercise 3

Solve for z: 2/(z+3) ≠ 2/3z.