Simple inequalities

Continuing on this path of solving equations with a variable, let’s take a look at solving simple inequalities.

At this point, you’re comfortable with solving for a variable on one or both sides of the equation, right? Honestly, solving simple inequalities isn’t much different. There’s one small detail you have to watch out for, but otherwise it’s the same process.

Let’s take a look at the following inequality:

3x + 5 < 26

Just like we did when we were solving for a variable on one side of an equation, we’re going to start by grouping all of our non-x terms together.  We’re going to subtract 5 from both sides, which will leave us with:

3x < 21

Now, we’ll divide both sides by 3 (again, we’re always working to keep our equation or inequality balanced) to get:

x < 7

It’s fairly straightforward, right? Well, let’s take a look at the next example.

4x + 4 > 6x + 8

We’ll start by grouping our x terms together. We’re going to subtract 6x from both sides. This gives us the following:

-2x + 4 > 8

Now we’ll subtract 4 from both sides to group our non-x terms together. The inequality now looks like this:

-2x > 4

We now divide both sides by -2 to get:

x < -2

Wait a minute! The inequality just changed direction! What gives?  Well, what gives is that whenever you multiply or divide both sides by a negative number, the inequality changes its direction. That’s the unique fact you have to remember when working with inequalities.