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## Checking solutions of two-variable inequalities

# 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 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.