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## Pre-algebra

### Course: Pre-algebra>Unit 7

Lesson 7: Intro to inequalities with variables

# Testing solutions to inequalities

Inequalities are like puzzles where you compare two things. You can solve these puzzles by trying different numbers and seeing which ones make the puzzle true. Some numbers will work for one puzzle but not for another. It's all about finding the right fit!

## Video transcript

- [Voiceover] We have two inequalities here, the first one says that x plus two is less than or equal to two x. This one over here in I guess this light-purple-mauve color, is three x plus four is greater than five x. Over here we have four numbers and what I want to do in this video is test whether any of these four numbers satisfy either of these inequalities. I encourage you to pause this video and try these numbers out, does zero satisfy this inequality? Does it satisfy this one? Does one satisfy this one? Does it satisfy that one? I encourage you to try these four numbers out on these two inequalities. Assuming you have tried that, let's work through this together. Let's say, if we try out zero on this inequality right over here, let's substitute x with zero. So, we'll have zero plus two needs to be less than or equal to two times zero. Is that true? Well, on the left hand side, this is two needs to be less than or equal to zero. Is that true, is two less than or equal to zero? No, two is larger than zero. So this is not going to be true, this does not satisfy the left hand side inequality, let's see if it satisfies this inequality over here. In order to satisfy it, three times zero plus four needs to be greater than five times zero. Well three times zero is just zero, five times zero is zero. So four needs to be greater than zero, which is true. So it does satisfy this inequality right over here so zero does satisfy this inequality. Let's try out one. To satisfy this one, one plus two needs to be less than or equal to two. One plus two is three, is three less than or equal than two? No, three is larger than two. This does not satisfy the left hand inequality. What about the right hand inequality right over here? Three times one plus four needs to be greater than five times one. So three times one is three, plus four. So seven needs to be greater than five, well that's true. Both zero and one satisfy three x plus four is greater than five x, neither of them satisfy x plus two is less than or equal to two x. Now let's go to the two. I know it's getting a little bit unaligned, but I'll just do it all in the same color so you can tell. Let's try out two here, two plus two needs to be less than or equal to two times two. Four needs to be less than or equal to four. Well four is equal to four and it just has to be less than or equal, so this satisfies. This satisfies this inequality. What about this purple inequality? Let's see, three times two plus four needs to be greater than five times two. Three times two is six plus four is ten, needs to be greater than 10. 10 is equal to 10, it's not greater than 10. It does not satisfy this inequality. If this was a greater than or equal to it would have satisfied but it's not. 10 is not greater than 10. It would satisfy greater than or equal to because 10 is equal to 10. So two satisfies the left hand one but not the right hand one. Let's try out five. Five plus two needs to be less than or equal to two times five, once again everywhere we see an x, we replace it with a five. Seven needs to be less than or equal to 10. Which is absolutely true, seven is less than 10. So it satisfies less than or equal to. Five satisfies this inequality and what you're probably noticing now is that an inequality can have many numbers that satisfy. In fact they sometimes will have nothing that satisfies it and sometimes they might have an infinite number of numbers that satisfy it and you see that right over here. We're just testing out a few numbers. For this left one, zero and one didn't work, two and five did work. This right one, zero and one worked, two didn't work. Let's see what five does. In order for five to satisfy it, three times x. Now we're gonna try x being five. Three times five plus four needs to be greater than five times five. Three times five is fifteen, fifteen plus four is nineteen. Nineteen is to be greater than 25, it is not. So five does not satisfy this inequality right over here. Anyway, hopefully you found that fun.