Question

Write an inequality for the graph,
please help me understand.

Answer 1

@jim_thompson5910 , could you help?

Answer 2

Do you remember which function has a V shape?

Answer 3

I do not, I don't remember anything lol. If you don't mind just explaining it a bit, then hopefully I catch on, if that's too much to ask. It's okay.

Answer 4

Lemme see if the drawing tool is working today, if not it might be difficult to go into detail.

Answer 5

Grr it's still broken >:O annoying.

Answer 6

Start with the function \(\Large\bf\sf y=x\)
Remember what that one looks like?
It's a straight line going upwards at a 45 degree angle, through the origin.

Answer 7

The absolute function: \(\Large\bf\sf y=|x|\) takes any negative x values and flips them up to make them positive instead.

Answer 8

https://www.desmos.com/calculator/ii4wfsbfcm

Answer 9

So the absolute function takes that red line on the left and flip it's upward to be positive instead.

The absolute gives us that nice v shape.

Answer 10

\[\Large\bf\sf y=|x|\]So we'll start with this.
We'll have to make some horizontal and vertical translations to get it to match up with our picture.

Answer 11

In the picture, the `vertex`, (middle point) has been shifted from (0,0) to what appears to be aboutttttt (5,1), does that sound about right? :o

Answer 12

I'm talking about the very base point of our icecream cone :O

Answer 13

Yeah it does Lol .

Answer 14

For our vertical translation we'll just add 1 to our function.\[\Large\bf\sf y=|x|+1\]That will move it up 1.
Do you remember how to deal with horizontal shifts? It's a little bit different than with vertical.

Answer 15

Yeah a little, it seems very familar I feel like I have it in my notes somewhere lol.

Answer 16

Fine fine fine I won't make you dig.
For horizontal shifts, we make the `opposite` adjustment.

So in this case we'll `subtract` 5 from x, since our vertex needs to move to positive 5.

Answer 17

\[\Large\bf\sf y=|x-5|+1\]

Answer 18

So there is function.
Hopefully that makes a bit of sense.
I know moving stuff around can be a bit tricky.


Now we need to deal with the shading.

Answer 19

You're awesome, Im writing that down now. So what's with the shading?

Answer 20

The shading let's us know that we'll need to write this an `inequality`, no longer using an equals sign.

The shading is telling us to include all of the values `larger` than the line we came up with.

Answer 21

Look back at our function, the right side represents the line.
We want the y values which are `larger` than the line.

We want y `to be larger than` the line.

Can you think of which inequality sign we might use? :O

Answer 22

umm

Answer 23

Didn't mean to post that
still thinking sorry lol

Answer 24

what about = ?

Answer 25

Hmm = is no good.
That will give us all of the y values that are equal to the line.
We want all of the y values that are "greater than" the line.
See how the area `above` the line is shaded?

Answer 26

Ohh >

Answer 27

Ah good! That will work!
\[\Large\bf\sf y\gt\text{the line}\]
\[\Large\bf\sf y>|x-5|+1\]

Now we have one more thing to deal with.

Answer 28

Should we use a strict inequality \(\Large\bf\sf \gt\) or the other one \(\Large\bf\sf \ge\) ?

Answer 29

\(\Large\bf\sf \gt\)
Strict means, we only want the values larger than the line.
Strict will exclude the values which fall ON the line.
When we exclude the values on the line, we draw a `dotted line` for our function.

\(\Large\bf\sf \ge\)
If we're `including` values along our line, we instead draw a `solid line`.
This symbol means, all of the values larger than ( the shaded region ), or equal to ( values along the line ).

Answer 30

So does our v shape have a dotted or solid line?

Answer 31

um, yes?

Answer 32

Don't let the dots inside the shaded area confuse you.
We don't care about the shaded part right now.
So the line part of our V, is it `dotted` or a `solid` line?

Answer 33

It's not a yes or no question silly! :3

Answer 34

Oh lol, gesh man only if you knew the day I had. This screen is giving me a headache lol.

Answer 35

XD

Answer 36

Hmm it looks like the V is a solid line in the picture, yes?
So we'll want to use the inequality that has the little equal line under it.
We want to `include` values along that line.
That's what the solid line is telling us.

Answer 37

So we'll want to use the \(\Large\bf\sf \ge\) symbol.
Not the \(\Large\bf\sf \gt\).

Answer 38

\[\Large\bf\sf y\ge|x-5|+1\]

Answer 39

Ohhh I see lol.

Answer 40

https://www.desmos.com/calculator/4tg2wtkgzl
Here's another look at it.
A nice graphing utility to check your work.

Answer 41

So that would be our final answer.
I know it's a bit tough, lot of little things to remember.

Answer 42

Yes, yes it is. Thanks alot

Answer 43

np :3