Checking solutions of two-variable inequalities
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 again, let me 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 for 3 and 5. So when x is 3, y is 5. Let's see if this actually gets satisfied. So we get 5 times 3. Let me color code it. So this is 5-- I didn't want to do it in that color-- 5 times 3 minus 3 times 5. Let's see if this is greater than or equal to 25. So 5 times 3 is 15. And then from that, we're going to subtract 15, and let's see if that is greater than or equal to 25. Put 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 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 is to be greater than or equal to 25. This is 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 26 is indeed 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 we have 5-- let me do this is a new color. So we have 5x-- 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, and 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 25/3. So this is now the expression or the inequality, y is less than or equal to 5/3 x minus 25/3. 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. So our y-intercept is negative 25/3. That's the same thing as negative 8 and 1/3. So that's 1, 2, 3, 4, 5, 6, 7, 8, and a little bit more than 8. So our y-intercept is negative 8 and 1/3 like that. And it has a slope of 5/3. So that means for every 3 it goes to the right, it rises 5. So it goes 1, 2, 3, it rises 5. 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 a y is equal to 5/3 x minus 25/3. But here we have an inequality. It's y is less than or equal to. So for any x, the y's that satisfy it are the y's that equals 5/3 x minus 25/3-- 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 the less than 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, 2-- it's 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 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.