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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to solve any kind of linear inequality problem where you only have to perform two steps. You will know in what order to perform each step and you will know just what kind of operation you need to do.

You are visiting John today who works for a computer company. John has invited you to dinner at a restaurant and then games at his house when he is done at work. You're excited for the night to begin so you ask John how much longer he needs to work. He tells you, 'Oh, less than 2 hours.' Since John also likes math a lot, he writes what he has said out on paper in math speak. He writes *x* < 2 hours.

This is a **linear inequality**, a linear mathematical statement with inequality signs instead of equal signs. So you wait for John to finish his work. John likes to play games to work the brain, so he now tells you that it will take 40 - *x* > 10 minutes to get to the restaurant where you guys are going to have dinner. You have a confused look on your face, so John tells you that if you want to find out how many minutes that really is, you will need to solve that problem for the variable *x*.

How do you solve these types of problems? Well, you want to get the *x* by itself and you want the *x* to be positive. So you need to perform some operations to make this happen. How do you know what operations to do?

Well, you first look to see if there is any addition or subtraction going on. If there is, then you go ahead and perform the opposite operation to move things around so that your *x* becomes positive and is by itself. So if you see addition, you perform subtraction and vice versa.

Second, you look to see if there is any multiplication or division connected with the variable. If there is, you perform the opposite operation to get the *x* by itself. So, if there is multiplication, you divide, and vice versa.

In this video lesson, we are covering only those linear inequalities that require just two steps to solve. Usually this involves performing either addition or subtraction and then following it up with either multiplication or division.

Let's take a look at solving the problem 40 - *x* > 10. This step requires two steps.

First, we see that a 40 is being added to our variable. We can subtract the 40 from both sides of our linear inequality. We get:

40 - *x* - 40 > 10 - 40

Which turns into:

*-x* > -30

Now, we see that there is a negative 1 being multiplied with the *x*. This tells us that we now need to divide by a -1 on both sides of our inequality. We get:

*-x*/-1 > -30/-1

Which turns into:

*x* < 30

It's important to note that if we multiply or divide by a negative number, then our inequality sign flips. See how we began with a greater than and ended up with a less than? So what does this answer tell you? It tells you that it will take less than 30 minutes to get to the restaurant. You're getting hungry, so that works out well.

On the way to the restaurant, John gives you two more problems to do just for the fun of it. He tells you it's good for your brain, that it keeps your brain young. The first problem he gives you is this:

3*x* - 2 < 10

You see that the *x* is being multiplied by 3 and a 2 is being subtracted from it. Since you are looking for addition or subtraction going on first, you tackle the 2 first. To make it go away, you need to add the 2 to both sides of this inequality. You get:

3*x* - 2 + 2 < 10 + 2

Which becomes:

3*x* < 12

Then you tackle any multiplication or division. There is multiplication by 3, so you divide by 3 on both sides of the inequality. You get:

3*x*/3 < 12/3

Which becomes:

*x* < 4

John smiles so that means you got the right answer.

John gives you another one.

*x*/-4 + 1 > 8

Here you see that the *x* is being divided by a negative 4 and a 1 is being added to it. You tackle the 1 first. You need to subtract 1 from both sides of the inequality first. You get:

*x*/-4 + 1 - 1 > 8 - 1

Which becomes:

*x*/-4 > 7

Now you can tackle the -4. Since this is division, you go ahead and multiply the -4 to both sides of the inequality. You get:

(*x*/-4)*-4 > 7*-4

Which becomes:

*x* < -28

Yes, you remember to flip the inequality sign since you are multiplying by a negative number.

John gives you a big smile. You got this one right too! Yay! And you guys have arrived at the restaurant. John says, since you got all those answers right, he is going to pay for you so go ahead and order whatever you want.

Let's review what we've learned. A **linear inequality** is a linear mathematical statement with inequality signs instead of equal signs. In this video lesson, we look at how to solve those linear inequalities that require only two steps to solve.

To solve these types of problems, you first look to see if there is any number that is being added to or subtracted from the variable. If there is, you perform the opposite operation to move the number over and away from the variable. If you see subtraction, you add and vice versa.

Next you look to see if there is any multiplication or division going on with the variable. If there is, you again perform the opposite operation. If you see multiplication, you divide and vice versa. If you are multiplying or dividing by a negative number, then you need to make sure that you flip your inequality sign.

Once you have finished this lesson, you should be able to:

- Identify a 2-step linear inequality
- Solve a 2-step linear inequality problem

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

- What is an Inequality? 7:09
- How to Graph 1- and 2-Variable Inequalities 7:59
- Set Notation, Compound Inequalities, and Systems of Inequalities 8:16
- Graphing Inequalities: Practice Problems 12:06
- How to Solve and Graph an Absolute Value Inequality 8:02
- Solving and Graphing Absolute Value Inequalities: Practice Problems 9:06
- Translating Math Sentences to Inequalities 5:36
- Solving One-Step Linear Inequalities 7:08
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