In this lesson, we're going to learn how to solve two-step inequalities our first example is three. X plus 14 is less than or equal to 20. So whenever we're solving any type of equation or inequality, it's helpful to be able to separate the left side from the right side so that we can see exactly what's going on. So I'm going to draw a line down through that Inequality sign so I can see both sides separately. And now I focus on the side that has the variable, which in this case is x, and X is on the left Side of this problem. And our goal is to get X by itself, so we also have the three and the 14 on that side. We need to figure out how to undo those numbers so that we can get X by Itself. Now, since we do have two numbers that we have to undo, we have to do this in two steps. And that's how we know that this is a two step problem. So how do we know where to start? To figure that out. We follow the Order of Operations, but in reverse. So instead of starting with parentheses and then exponents multiplication, division and addition and subtraction, we'll start from the bottom and work our way up. So we have here the three that's being multiplied by the x. So we do have multiplication and then we have plus 14, so we also have addition. And since we're following PEMDAS in reverse, we're going to start by undoing the addition. So we'll do the opposite of adding 14, which is subtracting 14, and we'll have to do the same thing on both sides. So we'll also subtract 14 from the right side. And now we'll see what we have left. Well the positive 14 -14 will cancel out that would equal zero So on that side, we only have three X. To bring down and on the right side we have 20 -14 which is Six and now we need to figure out what symbol goes in between. Well, since this step only involves subtracting a number, there's no sign change. So we can keep our inequality sign the same as a less than or equal to sign. Now we just need to undo the three in order to get X by itself. Three is being multiplied by X. So we'll have to divide by three and do the same thing on the other side. Three over three will cancel out and we'll have just X on that side and on the right side, six divided by three leaves us with two. This step involves dividing by a positive number, so that also requires no sign change. The sign is going to stay the same. So our solution is x is less than or equal to two, and we can always check our answer to make sure that it makes sense. So to do that, we start with our original problem and we figure out some number to substitute for x. Since our solution is x is less than or equal to two, we just need to pick any number that satisfies that. So I'm going to use the number one and plug that in for x. So three times one plus 14 is less than or equal to 20. And now I'll simplify this and see if I end up with inequality that works. So I have three times one becomes three, and then if I add the three plus 14, I'll get 17. So this becomes 17 is less than or equal to 20, which is true. So now I know that my inequality does make sense. I do have the correct symbol. My solution does make sense for this problem. Next, we have negative x over five minus six is greater than eight. Now, if you'll notice, we have a negative sign to the left of our fraction, which is x over five. When it comes to negative signs and fractions, you can leave that negative sign out to the left of the whole fraction, or you could place it in the numerator or denominator as well. So to help us work through this problem a little bit easier, I'm going to move that negative sign so that it's down in the denominator with the five. I could have also put it in the numerator with x. But our goal is to get x by itself. So if we put the negative sign up there, we would actually be giving ourselves another step to do so. Now that I have the negative sign down with the five, I'm going to go ahead and start working through this problem to get x by itself. So just like before, we have to follow the order of operations in reverse. Now let's see what we have here. We have x over negative five. Then a fraction represents division. So we do have division in this problem. And then we have minus six. So we also have subtraction. Since we're doing the order of operations in reverse, we'll start by undoing the subtraction step. So the opposite of subtracting six is to add six and do the same thing on the other side. See what we're left with? Our minus six and plus six will cancel out. So bring down the x over negative five, and on the right side, eight plus six gives us 14. Now, this step involved adding a positive number, so there's no sign change, and I'll keep our inequality as the greater than sign. Next, we need to undo that negative five that's in the denominator of that fraction under x. So since x is being divided by negative five, the opposite will be to multiply by negative five. And we'll do that on both sides. So on the left side, negative five over negative five cancel out. So we have x left, and on the right side we have 14 times negative five, which is negative 70. And this step involved multiplying by the negative five. Since we multiplied by a negative number, this time we do have a sign change. Remember, we change the sign. We flip it anytime we're multiplying or dividing by a negative number. So now, instead of having a greater than sign, we'll have a less than sign. Our solution is x is less than negative 70. And now we'll check to make sure that it makes sense. Start with our original problem. And we need to pick a number to plug in for X. Since our solution says X is less than negative 70, I just need to pick any number that's less than negative 70. I'm going to pick negative 80. So plug that in for X. So now I can simplify this by doing negative 80 divided by negative five. That gives me 16. And then 16 minus six leaves us with ten. So now we have ten is greater than eight, which is true. So since that's true, we know that our solution is correct. It does make sense.