Write and graph a linear inequality that represents th situation.

Question

you have $48 to spend on music and downloads. Each album download costs $5 and each movie download costs $8. Let x represent the number of albums and y the number of movies.

Answer 1

To start out, you have to write the inequality they are asking for. I'm assuming we can't spend any more than $48, but we don't have to spend all of $48. If it costs $5 for every album (x) and $8 for every movie (y), than we'd write an inequality:

 

5x + 8y ≤ 48 

 

This means we can spend a max of $48, but don't have to spend all of it. Now we have to write the equation in a way that makes it easier to graph (you could graph it now by doing an x/y chart, but you'll be rearranging these types of equations a lot, so you might as well get the practice in).

 

The ultimate goal is to get y by itself, so start by moving 5x over to the other side

 

8y ≤ 48 - 5x      divide both sides by 8

y ≤ 6 - 5/8x      

 

I like to write all my line equations in "y=mx+b" so we'd write

y≤ -5/8x +6

 

m = -5/8 and y-inter equals 6

 

now graph your line. Since we're graphing an inequality, we have to shade in the area of the graph that still answers the question. So pick a simple points (I always choose (0,0)) and plug it in and see if it works.

 

0 ≤ -5/8(0) + 6

0 ≤ 6

 

If this is true, you shade over the point (0,0)... is it true? 

YES! So shade from your line down towards the x-axis. These are the combinations of albums and video downloads you can purchase without busting your $48 budget.

 

If you have any more questions, need clarification, or want to see the graph (I can't upload any images here), I'd be happy to help you out.

Answer 2

The important parts of this problem are the units and the inequality (a dollar limit).
 
First, units.
      "Each album costs $5"   means   $5 per album,     written as    $5/album
         and if there are x albums downloaded, the $ cost is  ($5/album)*(x albums)   
                                   [note: albums cancel, leaving $]
      "each movie costs $8"   [do the same thing with y movies]
      "$48 to spend"    means   total costs <=  $48
 
The inequality is:
        $5x + $8y <=  $48
 
Then, to graph the inequality, select two points on the line defined by  5x + 8y = 48
The easy ones are when x=8 and when y=0,  these are
                     (8,1) and (0,6)

                    [see if you can determine why I picked these two points for 48]
         Two points determine a line.
Now, does a point on either side of the line make the inequality true or false.  Let's choose (0,0) because it is easy.
          0 + 0 <= 48    is true, so all points on that side of the line you just drew make the inequality true.