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Systems of Inequalities - Problem 3

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

Some inequalities do not have a solution. When two inequalities have parallel lines and the shaded areas do not overlap (i.e., the opposite areas are shaded), then the system has no solution. This means that there is no coordinate point that makes both inequalities true.

Here I have a system of inequalities that I'm going to graph and I'm glad because they are both already solved for y so it should be pretty straight forward.

For the first equation I'm going to use red, I'm going to start at my y intercept of -2 go up my fractional slope up 3 over 4 and make another dot, wait up 3 over 4. Got it there it is. Then I'm going to use a dashy line to connect them and think about shading.

I usually use the point (0,0) to test as long as it's not on the line. Is it true that 0 is greater than 3/4 times 0 take away 2? O yes, is bigger than -2 so I'm going to shade up here, okay. Then for my other line I'm going to have a similar process starting at -3, 1, 2, 3, I need to do the slope which is up three over four, there it is. Draw a dashy line to connect them and then think about shading.

Is it true that 0 is less than ¾ times 0 take away 3? A negative yeah, is 0 less than -3? No, that means don't shade towards (0,0) for the blue line shade out here. Wow check it out you guys, this graph is a little hard to see because I didn't use graph paper don't make that mistake at home. But you can tell if I had done this perfectly this would be two parallel lines. The thing that's interesting here is I have no solution region.

Since I have parallel lines I have two lines that have the same slope and I can see that from my original problem is slope as 3/4 and I shaded away from each other no where on this graphs does my shadings overlap. That means no coordinates pair, xy coordinate pair, would work in both of these inequalities. That's kind of a tricky problem that you might see. Look out when you have parallel lines you might shade like this and not have any solutions at all.

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