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ELM: CSU Math Study Guide16 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Linear inequalities may look intimidating, but they're really not much different than linear equations. In this lesson, we'll practice solving a variety of linear inequalities.

Greater than and less than. These two symbols can be quite controversial. It all depends on what you put on either side. What if I said hot sauce is greater than ketchup? Or, cats are less than dogs? Or, the Denver Broncos are greater than the New England Patriots? These are debatable points. I mean, I think they're all true, and I'd argue them passionately, but they're really just opinions.

But, what if I said 6 > 7? Well, that's just wrong. 6 < 7. And, so are other numbers, including 5, 4, 3... well, the list just goes on from there. There's no debate. And, in this lesson, we're going to practice handling these types of situations.

You've already seen linear equations like this: *x* - 2 = 0. You solve for *x*, and get *x* = 2. Our variable, *x*, has a single value that we can determine. That's straightforward but also a little dry. It's like finding out that hot sauce is hot sauce and not comparing it to any potentially inferior condiments.

But, then there are linear inequalities. A **linear inequality** is a linear expression that contains relational symbols. That means that instead of =, you'll see >, <, __>__, or __<__. So, instead of our variable standing in for a single value, it's standing in for a relational value, as in *x* > 2, where *x* is all values greater than 2.

Let's solve some basic linear inequalities, then try a few more complicated ones. Just as with linear equations, our goal is to isolate the variable on one side of the inequality sign.

First, what about this: *x* + 5 > 9. We treat this just like we would if we had *x* + 5 = 9. We subtract 5 from both sides. Now we have *x* > 4. On a number line, that would look like the image below, where *x* is all numbers larger than, but not equal to, 4.

Note what *x* + 5 > 9 looks like. Here, the phrase *'x* + 5' can be shown as all numbers greater than 9. So, all we did was take that 5 away, which shifted our line so it looks like the image above.

Here's another: 18 < 12 + *x*. Okay, again, get the variable alone. Subtract 12 from both sides, and we have 6 < *x*. We could flip that around to say *x* > 6. Just remember that if you do that, don't forget to flip the inequality sign! That one looks like this:

Here's one: *x* - 7 __<__ 1. Let's add 7 to both sides to get *x* __<__ 8. This graph looks like this:

Note that we fill in the circle around the 8 because *x* isn't just less than 8, it's less than or equal to 8. *X* could be all these values as well as 8 - no sense making 8 feel left out.

Let's do one more basic one: *x* + 11 __>__ 14. This time, we subtract 11 from both sides, which gives us x __>__ 3. If we graph that, we get this line:

Again, we have a solid circle because *x* is greater than or equal to 3.

That's enough of the basic ones. Those are all like saying chocolate ice cream is greater than vanilla. Pretty straightforward, right? Okay, maybe in some circles.

Here's a trickier one: 3*x* - 6 < 9. Let's start by adding 6 to both sides. Now we have 3*x* < 15. What do we do? Remember, it's just like a linear equation. If we had 3*x* = 15, we'd divide both sides by 3. We do the same thing here. That gives us *x* < 5. That's it!

Here's another kind we haven't seen before: 7*x* + 2 < 8 + 5*x*. It looks more complicated, but the principle of getting that variable all alone hasn't changed. First, let's subtract 2 from both sides. Now, let's move the 5*x* over by subtracting 5*x* from both sides. Now we have 2*x* < 6. Just like the last one, we divide. So, we have *x* < 3. And, we did it.

There is one key rule that makes solving linear inequalities different than solving linear equations: if you multiply or divide by a negative number, you must reverse the inequality.

Let's see this in action: -2*x* < 8. This time, we divide by negative 2. So, we flip the sign. Now we have *x* > -4.

Does this make any sense? Well, if *x* > -4, what could *x* be? How about 1? If *x* = 1, is -2*x* < 8? Let's see. It'd be -2(1), which is -2. Is -2 < 8? Yes. We could keep trying this with all of the infinite numbers greater than -4, or you could trust me that it will always work. I say trust me. Otherwise, this lesson will be really long.

Let's solve one more: -12*x* - 3 < 19 - *x*. Let's start by adding 3 to both sides. 19 becomes 22. Now let's add the *x* to both sides. -12*x* becomes -11*x*. Finally, we'll divide by -11. That means flipping the sign. And, 22 becomes -2. It's important to not forget that negative sign. We're flipping the sign, but otherwise the normal rules of solving equations still apply. So, we have *x* > -2.

In summary, solving a **linear inequality** is just like solving a linear equation. The big difference is that instead of your variable being equal to a single number, like *x* = 10, it's defined in relation to a number. We use four symbols: >, <, __>__ and __<__. When solving, we treat it as though the inequality sign were an equal sign. There's just one exception: If you're multiplying or dividing by a negative number, the sign gets flipped.

Upon completing this lesson, you should be able to:

- Define linear inequality
- List the symbols used in linear inequalities
- Demonstrate how to solve basic and advanced linear inequalities

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ELM: CSU Math Study Guide16 chapters | 140 lessons

- What is a Linear Equation? 7:28
- How to Write a Linear Equation 8:58
- Problem solving using Linear Equations 6:40
- Solving Linear Equations with Literal Coefficients 5:40
- Solving Linear Equations: Practice Problems 5:49
- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- Solving a System of Equations with Two Unknowns 6:15
- Solving Problems Involving Systems of Equations 8:07
- What is an Inequality? 7:09
- Solving Linear Inequalities: Practice Problems 6:37
- Go to ELM Test - Algebra: Linear Equations & Inequalities

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