Compound Inequalities 03:59 minutes

Video Transcript

Transcript Compound Inequalities

Jerry and Larry want to ride the rollercoaster, but they have a "little big" problem. What does this have to do with compound inequalities? Let me tell you.

Let’s look at the problem first: You are only allowed to ride the rollercoaster if you are at least 4 feet 7 inches in height or shorter than 6 foot 3. Jerry is shorter than 4 foot 7 while Larry is taller than 6 foot 3.

Compound Inequalities on a Number Line: Conjunction and Disjunction

Let’s have a look at the number line to see how this problem can be represented graphically. It's not that you have to be an exact height to ride a rollercoaster, but you do have to be taller than a certain height as well as shorter than a certain height.

So you can say the height is greater than or equal to 4 foot 7, which is represented by a filled circle. Also, it has to be less than 6 foot 3, which is represented by an empty circle.

We'll use x to represent height. You can write this as a compound inequality with AND. x is less than 4 foot AND x is also less than 6 foot 3.

If you don’t look at the people who are allowed to ride the rollercoaster but instead at the people who aren’t allowed, like Jerry and Larry, you have a compound inequality with OR; either x is more than 4 foot 7 or x is greater than or equal to 6 foot 3.

This is represented graphically by these parts. Be careful: If you flip the wording in the problem, you also have to switch the circles! This means an empty circle at 4 foot 7 and an filled circle at 6 food 3.

How to solve Compound Inequalities: Conjunction

Let's take a look at an example to figure out how to calculate compound inequalities: 4 < 2x < 10.

This is a compound inequality with AND. So you can split this into two inequalities and put the word AND in between, ending up with one inequality, 4 < 2x and the other inequality is 2x < 10.

Now you calculate each inequality separately. Since we have 2x in both equations, we should divide both sides of both inequalities by two; the signs stay the same.

You can bring the two solutions, 2 < x and x < 5, back into one compound ineqality by reversing the process earlier. Remove the AND and now line up the Xs to get: 2 < x < 5.

How to solve Compound Inequalities: Disjunction

For compound inequalities with OR, you already have two seperate inequalities to begin with. So you simply calculate each individually.

In order to isolate the Xs, you have to eliminate 1 in the equation on the left, and 2 in the equation on the right. So you add 1 and 2 to the left and right equations, respectively.

Now you have to divide the equations by the coefficients, or the numbers in front of the x, which is 3 in the first equation and 4 in the second.

Your solutions, x < 5 and x > 12, can be brought together in the end with the word OR.

Let’s get back to Jerry and Larry. They really want to ride the rollercoaster. Jerry has an idea. Jerry decided to give his shoes to Larry. Now their heights are just right.