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Set Operation: Intersection - Problem 2

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

The intersection of two linear inequalities. When we’re looking at this, we’re looking for where these two inequalities overlap. However at this current condition, it’s sort of hard to figure out what’s going on. So we’re going to solve each linear inequality as you would normally.

For this one, I want to isolate the x, add one to each side, -3x is greater than 6, divide by -3 to solve it out, x is compared to -2. Remember when we divide by a negative we have to flip that sign, so this actually becomes the opposite sign. We’re still dealing with the intersection. Now we want to solve this one out.

Add 3, 4x is less than 16, leaving us with x is less than 4. That can actually go over here. We’ve rewritten our two statements to now be much more easy to deal with. I always draw this out on a number line, so I can actually see what is going on.

x is less than -2, so we have -2. Open circle, we’re not equal to it, and we are going less than. X is less than 4. Again we have an open circle, and again heading down. I always do it so that I’m sort of above the number line so I can always see what’s going on.

Now we’re dealing with intersection. Intersection is where they both exist, so we need to see where they both are occurring. Looking over here; less than -2. We have both lines that are in our intersection. Between -2 and 4, just the x less than 4, so that’s not in the intersection. And over here there’s nothing, so obviously that can’t be in the intersection as well. What we’re really dealing with is just this region over here, negative infinity to 2.

We have to be careful about whether we include 2 or not. 2 is included in the x is less than 4. But it’s not included in x is less than -2. I put 2 down there, that should be a -2, shouldn’t it? So that should be a -2. Looking at our -2, it’s included in the x is less than 4, it’s not included here. So it can’t be in the intersection because it’s only in one of these inequalities. We can’t include it in our answer, so we’re left with negative infinity to -2, not including -2.

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