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Solving Compound Inequalities with an Intersection (AND)

In previous lessons, we saw how to solve individual inequalities, with grouping and graphing.

In this lesson, we're going to look at how to solve compound inequalities, which are basically two inequalities. With intersections, we're looking at the points where the two inequalities overlap. What does that mean?

Let's break it down with an example using graphing.

Say you have two inequalities:

$$\color{#8925AE}{x<8}$$ $$\color{#29A603}{x>4}$$

Let's graph the first inequality, the purple inequality, on a number line.

On the purple graph, we can see an open circle on $8$ because the inequality is simply less than, not less than or equal to. Because we're looking at values that are less than $8$, the purple arrow goes to the left.

Now, let's graph the second inequality, the green one.

This is a graph of $x>4$, with an open circle and the arrow going toward the right.

Since we want to find the solution of the two inequalities, we want to find the points at which the two inequalities meet. Notice that it's similar to solving systems of equations by graphing?

Let's take a look at a graph to see how this works out.

The orange section is the points at which the two inequalities intersection. These are the points that are both less than $8$ and greater than $4$. Kind of sounds similar to intersection of sets, doesn't it?

A good way to double check your work for simple problems is to think about it! List out the values that are greater than $4$ and less than $8$, exclusive:

$$5, 6, 7$$

That seems to match what we've graphed on the number line. So it seems we're correct!

Solve Compound Inequality for Intersection (AND)

To solve AND/Intersection compound inequalities, first solve each inequality separately. Then combine your inequalities together so that it takes the form $$\# < x < \#$$ $$\# \le x < \#$$ $$\# < x \le \#$$ $$\text{or }\# \le x \le \#$$

(Videos) Compound Inequalities

by mathman1024

This video by mathman1024 works through problems with compound inequalities.

$\large{\text{Summary}}$

There are two types of compound inequalities: union and intersection. A union is an "or". It puts solutions to both inequalities together. An intersection is an "and". It's the solution set that contains only the solutions common to both inequalities.

The second example he look at is $x>5\text{ and }x<13$. Here the word "and" signals us that we use an intersection. We graph by selecting the numbers that are solutions to both inequalities.

In red, we see the graph of $x>5$:

In blue, we see the graph of $x<13$:

Let's look at the overlap:

In purple, we see only the overlapping region of $x>5$ and $x<13$:

We write the following to show this interval: $$(5,13)$$