Question

Another word problem in comments, please help. :(

Answer 1

Part II: Inequalities

3. As the owner of a business, you must make sure that you spend no more than you have. Unfortunately, you often must plan how the business will spend its money before the money has come in.

Let us say that it is the beginning of the second quarter of the year. You must put together a budget for the rest of the year. You are about to learn how much money the business made during the first quarter, and you can use this information to project how much money the business will make during the rest of the year (to project data is to make a prediction about data you do not have based on data you do have). Even though you have not yet learned how much money the business made during the first quarter, you have good reason to believe that during the rest of the year, the business will bring in about five times as much money.

You must be sure that you plan to spend less money than you expect to come in. At the same time, you know that the business will require a minimum of $150,000 to continue to exist during that time. Whatever budget you plan will include at least that much.

Your budget will be based on these two restrictions. The amount of money you spend in the rest of the year must NOT be more than five times what the business made last quarter, AND it must be at least $150,000.

Your task is to do the following:
• Write these two restrictions as a system of inequalities relating the amount you will spend in the rest of the year to the amount the business brought in last quarter. As always, be clear about which is which.
• Describe the graph of this system in words, indicating whether full or broken lines are used for each inequality and where there is shading.
• Name at least one pair of values that is a solution to the system.

Remember to use <= and >= to indicate “is less than or equal to” and “is greater than or equal to,” respectively

Answer 2

yes, its big and very intimidating

Answer 3

There is a lot, but it boils down to this

We have two variables: money that you will spend in the rest of the year and money that has been earned last quarter


So why not let

x = amount of money earned last quarter

and

y = amount of money to spend for the rest of the year



We're told (explicitly) that "The amount of money you spend in the rest of the year must NOT be more than five times what the business made last quarter"

So this means....


y <= 5x


and we're told (again explicitly) that "it [the budget] must be at least $150,000"

So we can say


y >= 15000

Answer 4

The reason why I made x be the amount of money earned last quarter is because this amount dictates the amount we can spend for the rest of the year

Basically, the lower the amount the company earns in the first quarter means the lower the overall budget. This is NOT true the other way around. The future doesn't dictate the past.

Answer 5

Your task now is to graph this system

y <= 5x
y >= 150000

Answer 6

and describe what it looks like

Answer 7

hmm yeah that makes sense.. The only thing I came up with was 30,000 lol but I'm not sure if that is correct or where to go from there.. I'll try graphing what you said though

Answer 8

Now that I think about it, it might be easier to think of x and y in the thousands of dollars since the numbers are so high

Answer 9

So instead of saying

y >= 150000


say

y > 150


where it will be understood that y is in thousands of dollars. Make sure you write this somewhere so your teacher knows what you mean.

Answer 10

okay

Answer 11

So you can say

x = amount of money earned last quarter (in thousands of dollars)

and

y = amount of money to spend for the rest of the year (in thousands of dollars)

Answer 12

oops meant to say

y >= 150

Answer 13

lol okay :)

Answer 14

So do you know how to graph this?

Answer 15

when you say ">=" you mean \[\ge \]
right?
yes I am sort of good at graphing so I'll try

Answer 16

yes >= means greater than or equal to

Answer 17

btw at the bottom of your instructions, it states

"Remember to use <= and >= to indicate “is less than or equal to” and “is greater than or equal to,” respectively"

Answer 18

so i'd stick with <= and >= instead of using other symbols

just in case your teacher gets picky

Answer 19

Hah oh yeah :) silly me, sorry bout that lol

Answer 20

that's ok

Answer 21

yeah, I'll do that

Answer 22

those instructions were (probably unnecessarily) long..so i don't blame you

Answer 23

lol yeah, the length was pretty intimidating.. and I'm honestly not good at word problems.. I get really confused :/ but anything else is usually easy for me.

Answer 24

well a lot of it is just fluff and unneeded in my opinion since it boils down to something like that

Answer 25

but I guess it doesn't hurt to have you practice word problems (since they'll be most common)

Answer 26

so as for y <= 5x ...
it would be y <= 5/1x
so up 5 over 1?

Answer 27

yes

Answer 28

okay

Answer 29

hmm I don't understand the second one... would I go up 150 and over 1?

Answer 30

y >= 150

has the boundary line

y = 150

which is the same as

y = 0x + 150

Answer 31

hmm okay

Answer 32

almost forgot

Answer 33

so then would that just be a line?

Answer 34

on y

Answer 35

yes a horizontal line more specifically at y = 150

Answer 36

on the y-axis

Answer 37

so your graphing scale should be pretty big (say each tick mark is 50...which represents 50,000 dollars)

Answer 38

okay

Answer 39

hmm I'm a bit confused now.. what do I do after making these lines?

Answer 40

Now because y <= 5x, this means that you shade below the line y = 5x

Because y >= 150, we shade above the line y = 150


The overlapping shaded region is the final solution set

Answer 41

oops I shaded below 150

Answer 42

here's the graph of the two lines

Answer 43

cool, mine looks just like it :)

Answer 44

okay, and how exactly do I find that solution in that shaded region.. I always had trouble figuring that out

Answer 45

and here's the shaded region

Answer 46

That blue region is the result of overlapping the red region (for y <= 5x) seen in the graph below

Answer 47

and overlapping it with the green region shown below

Answer 48

I got that right :)

Answer 49

alright great

Answer 50

what now? that's it?

Answer 51

now "Describe the graph of this system in words, indicating whether full or broken lines are used for each inequality and where there is shading"


then "Name at least one pair of values that is a solution to the system"

Answer 52

okay that's fine, I can do that.. but how exactly do I find a pair of values?

Answer 53

any point that lies in the blue region will work. So can you name any points in that region?

Answer 54

hmm oh okay, cool :)
(100, 200)

Answer 55

yes that point works perfectly

Answer 56

thank you :)

Answer 57

(100,200) means that x = 100 and y = 200

So if the company made $100,000 in the first quarter, then it can budget $200,000 for the rest of the year

Answer 58

Any other point that lies in the blue region will have a similar interpretation

Answer 59

oh and I should clarify....it can budget this money legally (and stay within the restrictions)

Answer 60

lol :) thank you, this was actually pretty easy after you broke it down (sorry I'm so slow lol)

Answer 61

you're fine

Answer 62

it's a lot to sift through (esp if you haven't had much practice with it)

Answer 63

yeah it is... well thank you again :) deadline in less than 3 hours..1 project down and 3 more to go lol

Answer 64

lol you're on fire and i'm sure you'll make it all in time

Answer 65

I hope so :( my mind has been in a funk lately

Answer 66

I'm sure you'll do just great

Answer 67

your the one thats on fire lol not me :P

Answer 68

yeah I hope so, thank you

Answer 69

lol you have been too

Answer 70

np