Solving Inequalities Using All 4 Basic Operations

Hello, and welcome to this video on solving inequalities. In this video, we will discuss:

• What an inequality is, and
• How to solve inequalities using addition, subtraction, multiplication, and division

What is an Inequality?

When solving equations, you have two expressions that are equal to each other. When we look at inequalities, we are looking at two expressions that are “inequal” or unequal to each other, as the name suggests. This means that one equation will be larger than the other.

The four basic inequalities are: less than, greater than, less than or equal to, and greater than or equal to.

Solving Inequalities

When solving inequalities, you follow all the same steps as solving an equation, except for a special rule when it comes to multiplication and division. The main difference is that instead of writing an equal sign between the two expressions, you will write one of the four inequality symbols.

Example #1

Let’s first look at an inequality using addition.

 
If we are solving for $x$ by itself, we want to get rid of that 7 next to it, so we subtract 7 from both sides.

 
This gives us our answer:

 
It’s as simple as that!

Example #2

Now, I want you to try one on your own using subtraction.

 
First, we are going to add 3 to both sides.

 
Then we simplify.

Example #3

Now we come to multiplication and division.

Are you ready to find out what this special rule is that I was talking about earlier? When you multiply or divide by a negative number, you have to flip your sign the opposite direction. If you are multiplying or dividing by a positive number, don’t worry about this step.

Let’s look at an example:

 
To get $x$ by itself, we need to divide both sides by -4.

Remember, since we are dividing by -4, we have to flip our inequality sign

 
Let’s take a second to look at why this happens. What if I didn’t flip my sign? I would have $x>-3$. So let’s try plugging in 2, since 2 is greater than negative 3. If we plug in 2 for $x$, we get:

 
But we know that this isn’t true; -8 is not greater than 12.

Now look back at our correct answer, $x<-3$. Negative 20 is less than negative 3, so let’s plug this into our equation to check and see if it works.

 
That’s true! 80 is greater than 12. So just remember, when you multiply or divide by a negative number, you HAVE to flip the sign. Otherwise, your inequality will not be true.

Example #4

What if we had this inequality?

 
For this inequality, we need to multiply both sides by 3. When we do this, do we flip our sign? No, we don’t have to since we are multiplying by a positive number.

So we’ll multiply both sides by 3, then we get:

Example #5

I want you to try one more on your own. For this one, we are going to combine everything we’ve learned, so it will look a little more challenging, but you can do it. Just apply each step that we have talked about so far.

 
Pause this video and solve this inequality on your own, then see if your answer matches up with mine.

Think you’ve got it? Let’s see!

First, I’m going to add 7 to both sides of my equation.

This gives us:

 
Now, I have to subtract $2x$ from both sides.

 
Finally, I need to divide by -1 and flip my sign.

So our final answer is:

 
Now, notice with this inequality, you could have subtracted $x$ and subtracted 3 from both sides. This will give you the same answer, and you can avoid dividing by a negative. Sometimes there are multiple ways to solve an inequality or an equation, so be on the lookout for ways to make your life a little bit easier.

I hope this video on solving inequalities was helpful. Thanks for watching and happy studying!

Solving Inequalities Practice Questions