Double inequalities

We're asked to solve for x. And we have this compound inequality here, negative 16 is less than or equal to 3x plus 5, which is less than or equal to 20. And really, there's two ways to approach it, which are really the same way. And I'll do both of them. And I'll actually do both of them simultaneously. So one is to just solve this compound inequality all at once. And I'll just rewrite it. Negative 16 is less than or equal to 3x plus 5, which is less than or equal to 20. And the other way is to think of it as two separate inequalities, but both of them need to be true. So you could also view it as negative 16 has to be less than or equal to 3x plus 5. And 3x plus 5 needs to be less than or equal to 20. This statement and this statement are equivalent. This one may seem a little bit more familiar because we can independently solve each of these inequalities and just remember the "and." This one might seem a little less traditional because now we have three sides to the statement. We have three parts of this compound inequality. But what we can see is that we're actually going to solve it the exact same way. In any situation, we really just want to isolate the x on one side of the inequality, or in this case, one part of the compound inequality. Well, the best way to isolate this x right here is to first get rid of this positive 5 that's sitting in the middle. So let's subtract 5 from every part of this compound inequality. So I'm going to subtract 5 there, subtract 5 there, and subtract 5 over there. And so we get negative 16 minus 5 is negative 21, is less than or equal to 3x plus 5 minus 5 is 3x, which is less than or equal to 20 minus 5, which is 15. And we could essentially do the same thing here. If we want to isolate the 3x, we can subtract 5 from both sides. We get negative 21. Negative 21 is less than or equal to 3x. And we get, subtracting 5 from both sides. And notice, we're just subtracting 5 from every part of this compound inequality. We get 3x is less than or equal to 15. So this statement and this statement, once again, are the exact same thing. Now, going back here, if we want to isolate the x, we can divide by 3. And we have to do it to every part of the inequality. And since 3 is positive, we don't have to change the sign. So let's divide every part of this compound inequality by 3. You divide every part by 3. This is equivalent to dividing every part of each of these inequalities by 3. And then we get negative 21 divided by 3 is negative 7, is less than or equal to x, which is less than or equal to 15 divided by 3 is 5. You do it here. You get negative 7 is less than or equal to x, and x is less than or equal to 15/3, which is 5. This statement and this statement are completely equal. And we've solved for x. We've given you the solution set. And if we want to graph it on a number line, it would look like this. This is 0. This is 5. This is negative 7. Our solution set includes everything between negative 7 and 5, including negative 7 and 5. So we have to fill in the circles on negative 7 and positive 5. And it is everything in between. That's our solution set. And so we can verify that these work. You could try out a number that's well inside of our solution set, like 0. 3 times 0 is 0. So you're just left with 5 is greater than or equal to negative 16, which is true. And 5 is less than or equal to 20. Or negative 16 is less than or equal to 5, which is less than or equal to 20. So that works, and that makes sense. You could try 5. If you put 5 here, you get 3 times 5 plus 5-- well, that's just 20. Negative 16 is less than or equal to 20, which is less than or equal to 20. That works. Negative 7 should also work. 3 times negative 7 is negative 21, plus 5 is negative 16. So you get negative 16, which is less than or equal to negative 16, which is less than or equal to 20. And you could try other values. You could go outside of our solution set. Try something like 10. 10 should not work. And you see here, if you put 10 here, you get 3 times 10 plus 5 is 35. Negative 16 is less than or equal to 35, but 35 is not less than or equal to 20. And that's why 10 is not part of our solution set.