INSTRUCTOR: Now, what if we have an absolute value expression with an inequality? What do we need to do? Well, we still need to write two equations. The first one is going to remain the same. Now, for the second one, we need to do two things. We need to change the positive 5 to a negative 5, and we need to change the direction of the inequality. So that's important. Make sure you understand that. And then after that, just solve it. 

5 plus 1 is 6. And then let's divide by 3. So x is greater than 2. Now, for the other one, negative 5 plus 1 is negative 4. And if we divide it by 3, we're going to get a fraction. x is less than negative 4/3. Now, because we're dealing with inequalities, we need to plot the solution on a number line. 

So we need 2. Negative 4/3 is about negative 1.3, which is between negative 1 and 2. So x is greater than 2, which means that we have an open circle at 2 shaded towards the right. And x is less than negative 4/3, which is about negative 1.3. So it's an open circle shaded towards the left. 

Now, the solution using interval notation-- let's write it up here. It's going to be negative infinity to negative 4 over 3 and then union 2 to infinity. So here's the 2 to infinity part. And here's the negative infinity to negative 4/3. The open circle is at negative 4/3. Just keep that in mind. 

Let's try another example. Feel free to pause the video and work on it. Now, right now, we can't write two equations yet. We need to get the absolute value expression on one side of the equation first. So first, let's subtract both sides by 3. So what we're going to have is negative 2 times the absolute value of 2x minus 1 is less than or equal to negative 10. 

Next, let's divide both sides by negative 2. So this will change the direction of the inequality. Just keep that in mind. So the absolute value of 2x minus 1 is equal to or greater than positive 5. Now, since we have the absolute value expression on one side by itself with no other numbers outside of it, this is when we can write two equations. 2x minus 1 is greater than or equal to 5, and 2x minus 1 is less than or equal to negative 5. 

Don't forget to change the direction of the inequality. So now let's add 1. So 2x is greater than or equal to 6. And next, let's divide by 2. So x is equal to or greater than 3. Here, we're going to add 1. So 2x is less than or equal to negative 4. And then divide by 2. So x is less than or equal to negative 2. 

So we have x is less than or equal to negativ 2. And x is equal to a greater than 3. So here's 0. Here's negative 2. And here's 3. So x is equal to or greater than 3. So we have a closed circle at 3 shaded towards the right. x is equal to or less than negative 2, so we have a closed circle shaded towards left. So therefore, the solution is negative infinity to negative 2, with a bracket, union 3 to infinity.