# Linear algebra inequalities - please give hints

• Jun 15th 2013, 03:57 AM
Crystuls
Linear algebra inequalities - please give hints
Three friends decide to buy some fish and chips.
One piece of fish costs \$2.50
One scoop of chips costs \$2.00
The friends only have \$20 between them to spend.
They want at least one fish each and at least two scoops of chips between them.
Write down three inequalities involving the fish and chips purchase and list the possible purchases they could make.

What I have done so far:
I have added the costs of 3 fish (\$7.50), 2 scoops of chips (\$4.00) together and found they spent a total of \$11.50 out of their \$20.00
I am now finding more possible purchases they could make such as buying another 3 fish (1 for each of them) - that's \$7.50 which now comes to a total of \$19.00 out of their \$20.00

What I need help with:
I need help with finding and writing inequalities. What even is an inequality? I definitely need help since I don't even know what an inequality looks like, let alone how to write one.

Thanks guys!
• Jun 17th 2013, 08:33 PM
brosnan123
Re: Linear algebra inequalities - please give hints
Answer is: Inequality is just an approximate comparison. It is an algebraic relation showing that a quantity is greater than or less than another quantity.

Let amount x they spent on fish and amount y on scoop of chips

Inequalities are:

x + y < = (less than equal to) 20

3x > = (greater than equal to) 7.50

2y > = (greater than equal to) 4
• Jun 17th 2013, 11:12 PM
rmohais
Re: Linear algebra inequalities - please give hints
Hi Crystuls,

Once you've gained an intuitive understanding of what an inequality is, you could start playing around with inequalities in the xy-plane to see how they are represented visually.

For example with the inequality y > -3x + 4 is visualized by first drawing the line y = -3x + 4, and then shading the area above the line. The line itself is drawn using dashes as opposed to a full solid line.

Have a look at the examples on the Exdraft website using the link below. You can plug in your own numbers and visualize an inequality, and even visualize several of them at once.

After you get a grasp of how these things behave, it becomes easier to take text-based scenarios like the one you gave, and convert them into inequalities, then graph them, then draw conclusions.

Cheers,
Rosemarie
• Jun 17th 2013, 11:48 PM
Draco93
Re: Linear algebra inequalities - please give hints
Right now we are given the above circumstances as stated in the given question. It's been a while since I've done one of these since high school, I will see what I can do from memory.
Let f be the # of fish they can buy under these circumstances.
Let c be the # of chips they can buy under these circumstances.

3 people
1 x fish cost \$2.50
1 x chips cost \$2.00
\$20 budget
At least 1 fish/person and at least 2 chips between them

Therefore:
2.5f + 2c < = 20 -------(1)
f > = 3 --------(2)
c > = 2 ----------(3)

Plot these lines on a graph. The area enclosed by these 3 lines should give you the combination of all the possible purchases. For equation (1), just make c or f as the subject (i.e. isolate it on one side). Just look at the points where (c,f) are both integers because you can't have 3.2 scoops of chips. When I plotted my graph I used c to represent the x-axis and f to be the y-axis.

Here's the graph:
http://puu.sh/3in9Q.png
A possible combination could be (2,6). This is 2 chips and 6 fish. Hence, 2(2) + 6(2.5) = \$19
Another one is (6,3). 6 chips and 3 fish. \$19.5

• Jun 18th 2013, 01:28 AM
Prove It
Re: Linear algebra inequalities - please give hints
Quote:

Originally Posted by rmohais
Hi Crystuls,

Once you've gained an intuitive understanding of what an inequality is, you could start playing around with inequalities in the xy-plane to see how they are represented visually.

For example with the inequality y > -3x + 4 is visualized by first drawing the line y = -3x + 4, and then shading the area above the line. The line itself is drawn using dashes as opposed to a full solid line.

When dealing with linear programming problems, it is advantageous to shade the area that is NOT needed, as then the feasible region can be immediately read off as the region that has not been marked. So the inequality y > -3x + 4 should have a dotted line y = -3x + 4 and then shading the area BELOW the line.
• Jun 18th 2013, 03:18 AM
rmohais
Re: Linear algebra inequalities - please give hints
Hi Prove It,

Yes, that certainly beats trying to figure out which area has been shaded the most, which is sometimes tricky business. :-)

Cheers,
Rosemarie