Question

The graph below is represented by which system of inequalities?

A x + y > 4 and 2x − y < 10
B. x + y < 4 and 2x − y < 10
C. x + y > 4 and 2x − y > 10
D. x + y < 4 and 2x − y > 10

Answer 1

Are you familiar with inequalities?

Answer 2

well the graph kinda confuses me

Answer 3

We are interested in the green region.

Answer 4

We want to know what y values put us there with respect to x values

Answer 5

Let's take A x + y > 4 and 2x − y < 10

Answer 6

Okay

Answer 7

x+y>4 means y > 4 - x and 2x-y < 10 means 2x-10 < y or y > 2x -10
Does this make sense? If not, we'll need to slow down.

Answer 8

I dont fully understand 2x-y<10 means 2x-10<y

Answer 9

okay

Do you know what 2x-y=10 means? That's with an "=" sign. This means all x and y where this equation is true.

Answer 10

Oh ok so if you pug in numbers it will equal 10?

Answer 11

Bingo

Answer 12

yes, there are numbers that make this equation true, that is equal to 10.

But there are also some x and y values that make this equation "Less" than 10.

Answer 13

Okay

Answer 14

that's the solution of the inequality .. which means a region one one side of the line in this case..

Answer 15

Those are the x and y we are looking for.

Now, let's arrange this equation a little differently so we get y on one side

2x -y < 10
is the same as
2x - y -2x < 10 - 2x
I just subtracted -2x from both sides, it doesn't change the inequality
Can you see why I did this?

Answer 16

So it could become equal?

Answer 17

I subtracted 2x to get rid of the 2x on the left side. I'm trying to get "y" by itself:

2x -y - 2x < 10 - 2x
is now
-y < 10 -2x
See that?

Answer 18

Oh okay

Answer 19

yea i do

Answer 20

but now we have -y not "y" So what can I do?

Answer 21

remove the negative

Answer 22

Yeah! Let's do that.

If I divide by -1 on both sides, that make -y/-1 = y and on the right side (10 - 2x)/-1 becomes 2x -10.

So far so good?

Answer 23

Yes good

Answer 24

Okay.

It's very important to remember the following. When you divide an inequality by a negative number you need to change the direction of the inequality.

Let me explain.

if we had -3 < 3 (which is true) and divided by -1 on both sides, we would have 3 < -3, which is a false statement. Do you see that? That is why we need to change the direction of the inequality when dividing by a negative number.

So this is what we have now:

-y < 10 -2x
y > 2x - 10

Let me know if this doesn't make any sense.

Answer 25

um i dont understand sorry

Answer 26

i got confused in the middle there

Answer 27

So you see that, for example, -3 < 3 right?

Answer 28

That is -3 is less than 3

Answer 29

yes -3 is less than 3

Answer 30

Okay. Now if I divide both sides by -1 I have

-3 / -1 < 3 / -1
or
3 < -3

This is because -3 / -1 = 3 and 3 / -1 = -3

But 3 is NOT less than -3.

Answer 31

yes 3 isnt less then -3 so that equation is untrue right?

Answer 32

That's exactly the reason we need to switch the inequality whenever you divide by a negative number.

Since we divided by -1, we do that now:

-3 < 3
-3 / -1 > 3 / -1
3 > -3
Which is now true. 3 IS greater than -3.

NOT changing the direction of the inequality when dividing by a negative number is a VERY common mistake.

Answer 33

So always change direction when dividing but dont change when multiplying?

Answer 34

Yes, change with multiplying by a negative also.

Answer 35

okay

Answer 36

*when

Answer 37

Let's continue

Answer 38

For A:

x + y > 4 and 2x − y < 10

y > 4 -x (first inequality)
y > 2x - 10 (second inequality)

And we now know how to get these.

Answer 39

But what does it mean graphically?

Answer 40

What is the slope of y = -x + 4 and y = 2x - 10? Yes, I know I'm using "=", we are going to need this 'equation' to graph the 'inequality'

Answer 41

y=mx+b the slope of 2x-10 is 2

Answer 42

yeah!

Answer 43

and for the other?

Answer 44

its -x i believe?

Answer 45

-1, close

Now where does y = -x + 4 and y = 2x - 10 cross the y-axis? That is, what is the 'equation' equal to when x=0? This is called the y-intercept of the equation.

Answer 46

y = -x + 4
y = -(0) + 4 = ? This is the y-intercept for the 1st equation

y = 2x - 10
y = 2(0) - 10 = ? This is the y-intercept for the 2nd equation

Answer 47

-8

Answer 48

for the second

Answer 49

i guess the first equation is -4

Answer 50

So the 1st equation crosses the y-axis at y = 0 + 4 =4 and the second crosses the y-axis at y = 0 - 10 = -10.

So, y-intercepts are 4 and 10.

Look at the graphs. We have one line crossing the y-axis at 4 right?

Answer 51

yea

Answer 52

Now, look at the other line. Do you think it's possible for it to cross the y-axis at -10?

(I should have put -10 above for y-intercept, not 10).

Answer 53

possible?

Answer 54

yes i think so

Answer 55

yeah ok, let's check a few more things to be sure

Answer 56

Looking at the graph, the 1st equation should also cross the "x-axis" at 4. You see that?

Answer 57

no are we talking about both dashed lines?

Answer 58

yes. we are talking about the dashed lines

Answer 59

ok i see that

Answer 60

Now, if our 1st equation does NOT equal 4 when y =0, then option A is OUT. Let's check:

y=-x +4
0=-x+4, is x = 4 here?