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Constraining solutions of two-variable inequalities

Sal determines which x-values make the ordered pair (x,-7) a solution of 2x-7y<25, both algebraically and graphically.

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Video transcript

- [Voiceover] "Which x-values make the ordered pair "X, comma negative seven, a solution of the "following inequality?" The inequality is two X minus seven Y is less than 25. And so they give us some choices, and I encourage you to pause the video and see if you can figure it out on your own. All right, now let's work through it together. They're constraining that Y is going to be equal to negative seven. And so if we make that constraint, we can replace this Y with a negative seven. So we can rewrite the inequality as two X minus seven times negative seven, since we're constraining Y to be negative seven, is less than 25. And so this is going to be two X minus negative 49, or two X plus 49, is less than 25. Now if I just wanna isolate the X on one side, which we see for these inequalities up here, so we could subtract 49 from both sides. So subtracting 49 from both sides, we get two X is less than, let's see, 49 minus 25 would be positive 24, so this would be negative 24. Now to isolate the X, we just divide both sides by two and we're not gonna change the inequality, since we are multiplying or dividing by a positive value. Positive two. So this is going to be X is less than negative 12. And lucky for us, this is a choice. So as long as, if Y is equal to negative seven, as long as X is less than negative 12, we will satisfy this inequality. Let's do another one of these. And this one is a little bit more visual. So "Which Y-values make the ordered pair," so in the last one we constrained what Y was, and we figured out what X values would satisfy this inequality. Now we're going the other way around. We're constraining X and we're saying, what Y values would make the ordered pair true? Or make it a solution? "Which Y values make the ordered pair 5 comma Y "a solution of the inequality represented by the graph below?" So they didn't give it to us algebraically, they gave it to us visually. And so to be a solution, that means we have to be in this blue area. So this pair, so negative five, comma six. That would be a solution to the inequality being depicted. Something that sits exactly on the line, this would not be a solution, because notice the line, as you can see this lower boundary line, is a dashed line. If it was filled in, than anything on the line would be a solution. But since it's dashed, things on the line aren't solutions. It's only things that are above the line are going to be solutions. So let's see what they're asking us to think about. So they're saying, they're constraining X to be equal to five. So X equals five is everything, let me see if I can draw this. X equals five is everything on that line right over there. Now, if we assume that X equals five, we're gonna be someplace on this line, how do we have to constrain Y to make sure that we are in the solution? Well, we have to constrain Y so that we are above the line, for X equaling five, just to be clear. So we have to be above, our possible points are gonna be the ones once again, we're constraining X equals five, the possible points are the ones, that I'm showing in magenta. And actually I could keep going if I want to. So Y is going to have to be greater than, it can't be greater than or equal to seven, it has to be greater than seven. If it was greater than or equal to seven, we'd be including the point on the line, but I already talked about this being a dashed line so we don't want to include the points on the line. We only want to include the points above the line. So Y is going to be greater than seven which is this choice right over there. If X is equal to five, as long as Y is greater than seven, we are going to be in the solution set.