Question

How do I write an inequality for this graph?
http://i.imgur.com/VkxeA.gif

Answer 1

first you have to find the equation of the boundary line, which looks like an absolute value function.

Answer 2

How do I go about doing that?

Answer 3

well, we start with the parent function for the absolute value, which looks like...\[y=\left| x \right|\]

Answer 4

Have you seen this function before?

Answer 5

Yes, I recognize it.

Answer 6

After we identify the parent function then we have to translate the parent function to become the function that is depicted.

Answer 7

Do you know how to translate functions?

Answer 8

Not quite. I've only learned a bit about it.

Answer 9

The parent function of absolute value is graphed as The lowest point is (0,0)

Answer 10

If you look at the inequality function for this problem the lowest point has been moved compared to this graph.

Answer 11

what is the new lowest point?

Answer 12

(5, 1)

Answer 13

this means the function needs to be translated five to the right and one up.

we translate to point (a, b)
by replacing x by (x-a) and y by (y-b)

so for our function we have
\[y=\left| x \right|\]
becomes
\[y-b=\left| x-a \right|\]
Then solve for y.

Answer 14

Oh, I see. Thank you very much.

Answer 15

There is still more to do... next we have to figure out if the inequality is
<,>, <=, or >=

Answer 16

And how do we do that?

Answer 17

we have to look at the original diagram for this problem and check if the line is solid or dotted AND whether the shaded area is above or below the function.

Answer 18

if the line is solid then we use \[\le, or \ge \]
if the line is dotted then we use <, or >

Answer 19

Ah, so we use ≤or≥ in this instance.

Answer 20

yup

Answer 21

last thing is to choose which one of these you need.

Answer 22

And how do we go about that?

Answer 23

we just have to check which side of the function is shaded. Is it the top or the bottom.

Answer 24

The top.

Answer 25

that means that the shaded represents where y is greater or equal to the function that we came up with.

Answer 26

Alright, awesome. So what should our equation be looking like?

Answer 27

did you ever take the point you found as the lowest point of the function and plug it into
\[y-b=\left| x-a \right|\]

Answer 28

\[y=\left| x-a \right|+b\]

Answer 29

Oh, so where does the ≥ come in?