Question

Select the inequality that corresponds to the given graph.



4x - 3y > - 12

x + 4y > 4

4x - 2y < - 8

2x + 4y - 16

Answer 1

Because the dotted line goes from (-2,0) to (0,4) the slope of the line is 2. If, using the point (0,4), we make a linear equation using point-slope form, we get (y-4)=2(x-0).

y-4=2x
-4=2x-y
Multiply everything by 2
-8=4x-2y

That appears to correspond to the third inequality at first glance.

Answer 2

@Bbjunkie --> Which inequality symbols goes in this option? 2x + 4y - 16

Answer 3

I also refer to point-slope formula, where\[y=mx+b\]m is slope, also known as rise over run, and I see that, from point (-2,0) to point (0,4) you rise 4 and run 2.\[\frac{4}{2}=2\]Those are just two points I chose to look at..

b is y-intercept, which is the y-value at which the line hits the y-axis [when x=0]. I see that it is 4.

so\[y=mx+b\]applies as\[y=2x+4\]

That's not exactly what we're looking at, though. I could've said this from the begining, but it's intuitive to say it here. You know what that dividing line is now.

Since it's a dashed line, you knoww that it is not included in the shaded region. What's shaded is all that is greater than that line (to say it simply). All the y's you shade are actually greater than 2x+4 for any x. Thus\[y>2x+4\]

Answer 4

But, uhhh.. Since it's multiple choice you can either
(1) do all the above and algebraically manipulate it to see if it is equivalent to any of your choices
or
(2) rearrange the choices into y ? mx+b and then check to see which one looks right
or
(3) sketch all of the graphs, see which one is "it"

Answer 5

I'd go with option (2), for multiple choice!!
Remember that the directions of the arrows only change when multiplying or dividing by a negative number, and potential equality never changes.

Answer 6

Solve each option to get y > mx + b, and see which one matches up with the graph or my solution. It's there! Good luck!

Answer 7

@theEric --> Without a clarification of the fourth option, 2x + 4y - 16, I don't know that we can get a conclusive answer.

(0,4) and (-2,0) are the y-intercept and x-intercept, respectively, of the boundary line. I was checking those against the option inequalities.

Answer 8

Oh, sorry, Directrix!

You're totally right!

I said, "Solve each option to get y > mx + b", but you're right in that you can't do that for the last option, 2x + 4y - 16.

\[2x + 4y - 16\]contains no equality or inequality, so it's just an expression. It's like having a sentence with only a noun!

But the other three options can be solved.
It's not the first one, so I'll use it as an example.
There are multiple ways to go about it with algebra, and here's one (that contains the tricky multiplication or division by a negative number):

Start with\[ 4x - 3y > - 12\]
and subtract 4x from both sides\[4x-4x-3y>-12-4x\]\[-3y>-12-4x\]
and divide both sides by (-3)\[\frac{-3}{-3}y < \frac{-12}{-3}-\frac{4x}{-3}\]\[1y<4--\frac{4x}{3}\]\[y<4+\frac{4x}{3}\]Note the direction change of the inequality.
And mix stuff up.\[y<4 + \frac{4}{3}x\]\[y<\frac{4}{3}x+4\]\[y<mx+b\]

It's one of the middle options. Good luck! If you guess,you have a 50/50 chance!

Answer 9

@theEric We shouldn't have to guess. @BbJunkie needs to tell us what the fourth option is.

I was looking at x-and y-intercepts and working from that point of view.

Answer 10

@Directrix
I actually know the answer... I just didn't want to outright say it... I was hoping @Bbjunkie would be able to feel the success of figuring it out on Bbjunkie's own, with just a little help.

It is one of the middle two options - and the fourth option isn't correct unless it was a pretty bad typo. If they threw in an expression to give you that option, then Bbjunkie's job is to rule it out. And our's, too, as we help Bbjunkie.

Answer 11

For finding which of the two options are correct, you have to look at each one and find out if it creates a shaded graph like the one Bbjunkie provided us with.

Answer 12

Third option from the top.
\[4x - 2y < - 8\]

Subtract 4x from both sides\[4x-4x-2y<-8-4x\]\[-2y<-8-4x\]
Divide both sides by (-2). Since it's a negative, the inequality direction will change.\[\frac{-2}{-2}y>\frac{-8}{-2}-\frac{4x}{-2}\]\[1y>4+2x\]\[y>2x+4\]\[y>mx+b\]

So this will be a dotted line to show that that line, where y = 2x + 4, is not included in the shaded region; and it will have a slope of 2 and y-intercept of 4.