] Solving Compound Inequalities
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» » Solving Compound Inequalities

Solving Compound Inequalities


In our last lesson, you were introduced to compound inequalities. In this lesson, we are going to take the next step and begin solving compound inequalities.

Hopefully you have studied our lesson on solving regular inequalities. If so, I think you'll find this lesson to be pretty easy. The only tricky part at times is graphing the solutions. Let's take a look at a couple of examples together.


In this lesson, we are only going to focus on conjunction compound inequalities. In our next lesson, we will focus on disjunction compound inequalities.

Compound Inequalities - Example 1


Solve and graph the following inequality: 10 < 2x +8 < 16

Notice that this is a "conjunction" inequality. Conjunction inequalities usually include the word "and".

If we were to read this aloud, we would say, "10 is less than 2x + 8 AND 2x+8 is less than 16". Notice how the word "and" separates this into two inequalities?

There are actually two different ways to solve this inequality. This first method is to separate this into two different inequalities and solve each independently. The second method is to solve both parts at the same time. I will show you both ways and you can decide which is best for you.

Method 1: Solve Two Inequalities Independently

For this method, we will use the word "and" to write two different inequalities.

Take note of how the expression in the middle is used twice in order to write two different inequalities.

compound inequality example solving compound inequalities

You may be thinking, "Wow, that seems like a lot of work for one problem." I have to agree with you. This first method does require more work since you have to solve two different inequalities.

Method 2, which I will show you next, is going to require less work because you can solve both parts of the equation at the same time. This is the method that I prefer.



Method 2: Solving Both Parts at the Same Time


We are going to solve this same inequality a different way. Instead of separating this into two inequalities, we are going to leave it as one."

Remember my famous saying for solving equations, "Whatever you do to one side of the equation, you must do to the other side?"

Well, we are going to change that saying just a little to, "Whatever you do to one side of the equation, you must do to ALL sides of the equation."

Notice that I just changed the word "other" side to "all" sides. Why? Because we are going to think of this inequality as having THREE sides. Take a look...


conjunction inequalities

So, whatever you do side 1, you must also do to sides 2 and 3. This of it this way and you'll have no problems. Ready to solve?


solving compound inequalities




Remember, these two examples are two different ways to solve the same problem. You get to choose which way works best for you!

We are going to look at one more problem together. Have you been wondering about the rule that states, "If you multiply or divide by a negative number then you should reverse your inequality sign"?

Well, that rules still applies. Take a look with me...


Compound Inequalities - Example 2


Solve and graph: -12 < -4x +4 < 16

solving compound inequalities




Hopefully you have a better understanding of solving conjunction compound inequalities. Just remember to follow the same rules as for regular inequalities, but now you are solving two inequalities within one problem.

In the next lesson, you will solve disjunction compound inequalities. These are the inequalities with the word "or".




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