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Systems of Inequalities - Problem 4

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Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

To solve a word problem using a system of inequalities, start by using the information from the problem to set up two inequalities that model the problem. Make sure you properly identify which is the independent (x) variable and which is the dependent (y) variable. Solve for y. Next, graph each inequality by first plotting the y-intercept (the constant) and then from there, using the slope (the coefficient of the x variable) to plot a second point. If the inequality is less than or equal to, or greater than or equal to, then draw a solid line. If the inequality is less than or greater than (but not equal to), draw a dotted line. To determine the region that should be shaded, choose a point anywhere on the coordinate plane and substitute the values in the inequality. If the inequality is true, then shade the region where the point was taken from. If the inequality if false, then shade the region on the other side of the inequality line. Take the same steps with the other inequality of the system. The area where the shaded regions of the two inequalities overlap represents all the possible solutions. Keep in mind that the solution has to make sense for the problem. For example, If the problem is asking about quantities, time or money, then the solution can't be negative.

All right guys let's tackle a system of inequalities word problem. You buy ground beef and ground turkey in bulk to make different kinds of chilly at your restaurant. Yeah chilly ground beef costs a $1.50 a pound and ground turkey costs $2.50 a pound. You want to spend no more than 9.50 total and you need at least 4 pounds of meat. Write a system of inequalities, graph the system to show all possible solutions.

Okay here we go, let's talk system. I'm going let y be beef, y be pounds of ground beef and x be pounds of ground turkey. So here's what I know, I know I need at least 4 pounds so however much beef I get plus however much turkey I get has to be 4 pounds or more so I'm going to write it like that. That's going to be the first equation in my system of inequalities.

Now I need to deal with the costs stuff. The amount I'm going to spend on beef is going to be $1.50 times y, right? I have to pay a $1.50 for every pound I buy and y represents the number of pounds. For turkey I'm going to spend $2.50 for every pound of x that needs to be no more than $9.50 so that has to be 9.50 or less. This is my system of inequalities that represents how much I but and how much I spend.

Now I'm ready to do my graph to show all possible solutions. I'm only going to make a graph in the first quadrant because I only have positive values. I'm not using negative numbers in this situation. For my x value I'm going to go I'm not sure how far yet, my y value I'm not sure how far I'm going to go yet either but when I'm doing with slash is I'm only going up to about 4 on each one because 4 is my highest quantity.

Let's go through and graph this line, I'm going to rewrite it in slope intercept form by subtracting x from both sides. Y is greater than or equal to -x plus 4. 1, 2, 3, 4 I'm going to use -1 for my slope down one over one, down one over one you guys always use graph paper so yours will be a lot more -precise than mine.

There we go that's my first line let's look at the second line. I want to solve this guy for y and right now I have this $1.50 business going on. In order to eliminate that $1.50 I'm going to go through and divide each one of these quantities by 1.5. That way I'll have y all by itself. So let me grab my calculator and I'm going to have, come on marker, there we go. Y plus whatever 2.50 divided by 1.50 is which is 1.66x and then its going to be less than or equal to whatever 9.50 divided by 1.50 is and I get 6.3.

By the way if you were to use fractions instead of decimals here you would be more precise. But for my purposes so I'm just doing an approximate graph I'm going to use it like this. I still need to solve this for y meaning to get y all by itself 1.66x plus 6.33 and I'm going to have to extend my graph a little bit so it fits.

Pardon me when I erase this equation you got that written down? Okay. I'm going to erase that and make my graph a little bit taller than the y axis.4, 5, 6 dollars and 33 cents above there. Starting at my y intercept I need to go down 1.6 boxes and over one. Go down 1.6 boxes and over one. Again this is a really rough sketch go down one point six boxes about like that and over one.

If I can get a few good points on there I'll be able to connect and get a sense of where my line goes. I'm doing a solid line because this is the less than or equal to sign. Just like I had a greater than and or equal to sign.

Okay when I did my first graph I forgot to do shading so let's go back to this guy and shade it appropriately. Let's test the point (0,0). Is it true that o is bigger than 4? No.

That means for this first line I graphed I don't want to shade towards 00 I want to shade away from (0,0) or above that line. For the second inequality I'm going to try 0,0 and I'll see that 0 is yes less than 6.33.That means for the second equation I do want to shade towards 0 I'm going to shade below it.

My solution region is where the two shadings overlap. It's in this little triangle right here. This means that any point in there represents how much turkey and beef I could buy.

Remember how in the beginning I called x a turkey number? And I called y my beef number or my quantity of turkey and beef? What that means is that if I used one pound of turkey and 1, 2, 3, 4 pounds of beef I'd be right in my solution region.

That's a possible amount I could buy. I could also buy 3/4 pounds of turkey. And 1, 2, 3, 4 pounds of beef that would work also because its in my solution region. Any amount of pounds of beef and turkey even if they are fractional would give me solutions to these system of inequalities as long as the points fall within the that solution region.

So this was a big long intimidating word problems and I had to use lots of different Math skills but it's really applicable to real life especially if you are someone who goes into marketing in your future this'll still help you keep track of how much you can spend on different items for your store or your business or whatever.

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