Hey, everyone. So we've already learned how to solve linear equations like 2x − 6 = 0 by finding some value for x that we can plug back into our equation to make a true statement. But now you're going to start to see problems that instead of 2x − 6 = 0 have 2x − 6 ≤ 0. So there I have an inequality symbol instead of an equal sign and this is called a linear inequality. So linear inequalities are literally just linear equations but with an inequality symbol instead of an equal sign. Now I know what you're thinking, why are you taking something that I already know how to do and changing it? But don't worry, everything that we know about solving linear equations can be used to solve linear inequalities. We're just going to see a slightly different solution and I'm going to walk you through everything that you need to know about inequalities. So let's go ahead and get started. Like I said with linear equations, we were looking for some value of x and we could just move some numbers over and end up with a solution. Now let's go ahead and start solving our linear inequality the same exact way we did our linear equation. So if I have 2x − 6 ≤ 0, I can start by moving my 6 to the other side the same way I would with a linear equation. So if I just add 6 to both sides, that will cancel, leaving me with 2 x ≤ 6. Now that I'm here, I can isolate x by dividing by 2, again the same way I would with a linear equation. Canceling that 2 out and leaving me with x ≤ 3. So I actually still ended up with a 3 there. Now instead of being equal to 3 it is less than or equal to 3. So that's the difference that we're going to see here.
So let's take a look at one more inequality and just change something slightly. Here I had 2x − 6 ≤ 0 and over here I have -2x − 6 ≤ 0. So let's see what happens here. Well, I start the same way. I can go ahead and move my 6 over, leaving me with -2x ≤ 6. And now I need to go ahead and divide both sides by negative 2 to isolate it. So that will cancel, leaving me with x ≤ -3, and this is my solution. But if this is my solution, then that means that I should be able to plug any value that is less than or equal to negative 3 back into my original inequality and get a correct statement.
So let's see if that works here. So some number less than or equal to negative 3, if I try x = -4, that should work. So plugging that back into my original inequality, I have -2 × -4 − 6 ≤ 0. So -2 × -4 is positive 8, minus 6 is less than or equal to 0. 8 minus 6 is just 2, so 2 is less than or equal to 0. But 2 is definitely not less than or equal to 0, so something is wrong here and this actually is not my solution at all. So the reason why that's not my solution is because anytime that we'd multiply or divide by a negative number, I actually need to go ahead and flip my inequality symbol. So let's see what happens when we do that instead. So we're going to start again with this -2x ≤ 6. And then when I divide both sides by this negative 2, I'm still cancelling it out and left with x, but I'm going to take my inequality symbol and flip the direction it's facing. So instead of a less than or equal to sign, I'm going to use a greater than or equal to sign. And then I have 6 divided by negative 2, which gives me negative 3. So if this is my solution and this worked here, that means that any number greater than or equal to negative 3, I should be able to plug it. So if I take x = 0 and plug that in, that should hopefully work. Let's go ahead and give that a try. So -2 × 0 − 6 ≤ 0. -2 × 0 just goes away because anything times 0 is 0 and I'm left with negative 6 is less than or equal to 0. Now this is definitely a true statement. Negative 6 is less than or equal to 0. So this is my solution and x is greater than or equal to negative 3.
So let's talk about how we want to express these solutions. Well, whenever we were using doing linear equations, we just had one single value. We had something like x is equal to 3. So if I were to graph that, it would literally just be a single point on my graph because my solution is just a single value. But whenever we're dealing with linear inequalities, I don't have that x is equal to 0, I have that x is less than or equal to 0. And that's because I'm no longer dealing with a single value, but I am instead actually dealing with a whole range of values. So in order to graph that, I still have that same end point 3. And I know that since it's less than or equal to, that means I need to have a closed circle. So I'm going to draw a closed circle at my end point 3. And if x is anything less than or equal to 3, that means it could be anything to the left of that 3 all the way to negative infinity. And remember, we're also going to want to write these in interval notation. So in interval notation, remember that whenever we have an infinity it always gets a parenthesis. But should I enclose that 3 in a square bracket or parentheses? Well, again, since I have that less than or equal to sign, that means that 3 needs to get a square bracket because it is included in my interval. So let's look at our final example over here and rewrite our solution on a graph in interval notation. So since I have x is greater than or equal to negative 3, I still have my endpoint at negative 3 with a closed circle. Now since x can be anything greater than or equal to that negative 3, it's going to go all the way to the right up to infinity and forever. So if we write that in interval notation, my negative 3 is again going to get a square bracket here because I have a less than or equal to sign and not just a less than or a greater than sign. So then I have infinity which is of course going to get enclosed in parentheses and I'm done here. So that's all you need to know about solving linear inequalities. Let's get some more practice.