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# Testing solutions to inequalities

CCSS.Math:

## Video transcript

determine whether the ordered pairs 3 comma 5 and 1 comma negative 7 are solutions to the inequality 5x minus 3y is greater than or equal to 25 so we can literally just try each of these ordered pairs we could try what happens when X is equal to 3 and Y is equal to 5 in this inequality and see if it satisfies it and then we could try it for 1 and negative 7 so let's do that first let's do it first 4 3 & 5 so when X is 3 y is 5 let's see if this actually gets satisfied so we get 5 times 3 5 times 3 let me color code it so this is 5 I did want to do in that color 5 times 3 5 times 3 minus 3 times 5 minus 3 times 5 let's see if this is greater than if this is greater than or equal to 25 so 5 times 3 is 15 and then from that we're going to subtract we're going to subtract 15 and let's see if that is greater than or equal to 25 but that question mark there because we don't know and 15 minus 15 that is 0 so we get the expression 0 is greater than or equal to 25 this is not true 0 is less than 25 so this is not true this is not true so this ordered pair is not a solution to the inequality so this is not not a solution you put in X is 3 y is 5 you get 0 is greater than or equal to 25 which is absolutely not true now let's try it with 1 and negative 7 so we have 5 times 1 minus 3 times negative 7 needs to be greater than or equal to 25 5 times 1 is 5 and then minus 3 times negative 7 is negative 21 so it becomes minus negative 21 used to be greater than or equal to 25 this the same thing as 5 plus 21 subtracting a negative same thing as adding the positive is greater than or equal to 25 and 5 plus 21 is 20 6 is indeed greater than greater than or equal to 25 so this works out so this is a solution and just to see if we can visualize this a little bit better I'm going to graph this inequality I'm not going to show you exactly how I do it this time but I'm going to show you where these points lie relative to this solution so first let's put this in point so if we have so we have five we just in a new color so we have 5 X that's not a new color having trouble switching colors today we have 5x minus 3y is greater than or equal to 25 let me write this inequality in kind of our slope-intercept form so this would be the same thing if we subtract 5x from both sides we get negative 3y is greater than or equal to negative 5x plus 25 I just subtracted 5x from both sides so that gets eliminated you have a negative 5x over here now let's divide both sides of this equation or I should say this inequality by negative 3 and when you divide both sides of an inequality by a negative number multiply or divide by a negative number it swaps the inequality so if you divide both sides by negative 3 you get Y is less than or equal to negative 5 divided by negative 3 is 5 over 3x and then 25 divided by negative 3 is minus minus 25 over 3 so if I were to graph this so this is now the expression or the inequality Y is less than or equal to 5 3 X minus 25 over 3 so if I wanted to graph this I'll try to draw a relatively rough graph here but really just so that we can visualize this really so we can visualize it so our y-intercept is negative 25 over 3 that's the same thing as negative 8 and one-third so that's 1 2 3 4 5 6 7 8 and a little bit more than 8 so our y-intercept is negative eight and a third like that and has a slope of five thirds so that means for every three it goes to the right it rises five so it goes one two three it rises five it rises five so the line is going to look something like this I'm drawing a very rough version of it so the line will look something like that this line over here that's if it was y is equal to 5/3 X minus 25 over 3 but here we have an inequality it's y is less than or equal to so for any for any X the Y's that satisfy it are the Y that equals 5/3 X minus 25 over 3 that would be on the line so it would be that point there and all the Y's less than it so the solution is this whole area right over here since it's less than or equal to we can include the line the equal to allows us to include the line and less then tells us we're going to go below the line and we can verify that by looking at these two points over here we saw that 3 comma 5 is not part of the solution so 3 comma 5 is 1 to 2 right about there and then up 5 so 3 comma 5 is right above here it's in this region above the line and notice not part of the solution and then 1 comma negative 7 1 comma negative 7 is going to be right over here it's almost on the line so 1 comma negative 7 is going to be right over there but it is at least within this solution area so hopefully that gives you a little bit more sense of how to visualize these things and we'll cover this in more detail in future videos