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# Constraining solutions of two-variable inequalities

## Video transcript

which X values make the ordered pair X comma negative 7 a solution of the following inequality and the inequality is 2x minus 7 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 7 and so if we make that constraint we can replace this Y with a negative 7 so we can rewrite the inequality as 2x minus 7 times negative 7 since since we're constraining y to be negative 7 is less than 25 and so this is going to be 2x minus negative 49 or 2x plus 49 is less than 25 now we if I just want to 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 2x is less than what is this 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 2 and we're not going to change the inequality since we are multiplying or dividing by a positive value positive 2 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 7 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 the last one we we we constrained what Y was and we figured out what X values would would would satisfy this the 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 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 this pair so X so negative 5 negative 5 comma 6 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 this leg is you say this lower boundary line is a dashed line if it was filled in then anything on the line would be a solution but since it's dashed things on the line aren't solution so it's only things that are above the line are going to be solutions so let's see what they're 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 on that line right over there now if we assume that x equals five we're going to 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 why we have to constrain Y so that we are above the line for X equaling negative 5 for X equaling 5 just to be clear so we have to be above so these are possible let me do that a little bit our possible our possible points are going to be the ones once again we're constraining x equals 5 the possible points are the ones the ones that are that are I'm showing in magenta and actually I could keep going if I want to so Y is going to have to be greater then it can't be greater than or equal to 7 has to be greater than 7 if it was greater than or equal to 7 would be including the point on the line now you 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 Y is going to be greater than 7 which is this choice right over there if X is equal to 5 as long as Y is greater than 7 we are going to be in the solution set