Question

Which of the following inequalities matches the graph?

Answer 1

Can you determine the equation of the line that separates the two regions?

Answer 2

well it passes through 2 and -6

Answer 3

That's a start. What the slope be?

Answer 4

If it passes 2 when x = 2, y should equal 0 and that is not the case of any of the listed inequalities..

Answer 5

It actually goes close to, but not through, 2 and -6. We'll get the right slope, however...

Answer 6

so is it D?

Answer 7

Come on, I'm showing you how to systematically work these problems, including the cases where you don't have a menu of answers to choose from.

Answer 8

im open to learn im not just about getting the answers

Answer 9

So, the slope of the line is going to be what? It looks (to you) like it goes through (0,-6) and (2,0). It doesn't, but those numbers will give us the right slope anyhow. What is the slope of a line that passes through those two points?

Answer 10

Going from (0,-6) to (2,0), we go 2 units to the right and 6 units up, so our slope is 6/2 = 3. Agreed?

Answer 11

i agree with that

Answer 12

If you prefer the more formal description: \[(x_1,y_1) = (0,-6),\,(x_2,y_2) = (2,0)\]\[m = \frac{y_2-y_1}{x_2-x_1} = \frac{0-(-6)}{2-0} = \frac{6}{2} = 3\]

So our line dividing the two regions is of the form \[y = 3x + b\]

I think if you look more closely at the admittedly fuzzy graph, you'll see that the line actually passes through (0,-5).

Answer 13

But in any case, we have an inequality with a separating line that is dashed. Do you recall when we use solid lines and when we use dashed lines for the boundary in an inequality?

Answer 14

Solid lines are part of the solution set.
Dotted or dashed lines are borders of the solution
set but are not part of the solution set.

Answer 15

Okay. We have a dashed line, so the boundary or border is not part of the solution set. Does that mean we use \(<>\) or \(\le\ge\) for our inequality?

Answer 16

<>

Answer 17

We use a \(<\) or \(>\) here. Dashed line means no equals in the inequality symbol.

Next, we need to figure out which way the inequality points. To do this, pick a point on the graph that you can easily determine whether it satisfies the inequality. (0,0) is a good choice here. (0,0) is in the shaded region, so it satisfies the inequality. At any given point of x, the shaded region is where y > 3x + the y intercept of that line. Looks like the line passes through (0,-5) so the y-intercept is -5. Our line is y = 3x-5, and our inequality is y > 3x-5 because 0 > 3(0)-5. That is not one of our answer choices, so D is our answer.

Answer 18

Thank you so much! i understand a little better now

Answer 19

Not my best description, I'm afraid — a little too much blood in my coffee stream this morning :-)

Answer 20

But the general outline is okay:

1) find the equation of the line that makes up the border between the regions
2) convert it to an inequality of the right type by evaluating a convenient test point and choosing the right inequality symbol
3) whatever else is needed to finish the problem at hand

Answer 21

Okay, this one has a bit more clearly visible x and y intercepts!

What are they?

Answer 22

it looks like -2 and 3

Answer 23

yes, (-2,0) and (0,3). What is the slope of the line through those points?

Answer 24

y = 1.5x + 3

Answer 25

Yes. And do we have a solid or dashed border this time?

Answer 26

looks like a solid

Answer 27

Agreed. So what is our finished inequality?

Answer 28

i dont know :(

Answer 29

Let's test a point again. (0,0) is in the shaded region, and we have two choices:

\[y \le 1.5x + 3\]\[y \ge 1.5x+3\]Agreed so far?

Answer 30

yes agreed

Answer 31

Okay, which of those two equations is true when x = 0 and y = 0?

Answer 32

the second one

Answer 33

Are you sure?

\[0 \ge 1.5(0) + 3\]\[0\ge3\]???

Answer 34

ok lol the first one

Answer 35

Yes, the first one is correct. Now, because you converted a fraction to a decimal, your work is complicated by the fact that your equation doesn't look identical. Usually better to keep fractions as fractions until you need decimal numbers, in my experience.

\[y \ge \frac{3}{2}x+3\]is the equivalent of our inequality.

Answer 36

what's the equation of the border?

Answer 37

y = 0x + 0 ?

Answer 38

\[y = 0x+0 \]is the same as \[y = 0\]which is a horizontal line that sits on top of the x-axis. Is that the border here?

Answer 39

Isn't your border a line consisting of points with the same x value, but different y values?

Answer 40

yes

Answer 41

So, isn't that going to look like \(x = k\) for \(k\) being some constant?

Answer 42

y gets to be any number you choose, so long as \(x = k\). That's a vertical line at \(x = k\).

Answer 43

It has an undefined slope (and is not a function) because there are multiple points with the same x value. If you tried to compute the slope, you'd have a division by 0.

Answer 44

so it looks like its b

Answer 45

No :-(

The equation of the dividing line is x=2, is it not?