Solving Two-Step Linear Inequalities
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.
Linear Inequalities
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.
Solving Two-Step Inequalities
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.
Example 1
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:
3x - 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:
3x - 2 + 2 < 10 + 2
Which becomes:
3x < 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:
3x/3 < 12/3
Which becomes:
x < 4
John smiles so that means you got the right answer.