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Set Operation: Intersection - Problem 1

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!

Intersection of two linear inequalities. Here we have the intersection of these two statements. What we really want to do, is figure out what each of these statements are saying independently, and then piece them together. This just a linear inequality, we solve it as we would anything else.

To solve this out, subtract 2, we end up with x is less than or equal to 5.Still dealing with the union, solve this one out, subtract 2 on the other side. X is greater than or equal to -6.

Whenever I solve a union or an intersection, either one, I always make a number line. So make a number line and then plot both of these regions independently. X is less than or equal to 5. We have the number 5, and we have a closed circle because we are including it and it’s everything less than. I always do above the number line, so I can see exactly what’s happening and I don’t have anything just on top of each other. X is greater than, equal to -6. So here a -6, closed circle and that’s everything greater.

Now we’re looking for the intersection; remember intersection is where they both exist. Where they overlap. We want to see where both regions are represented. Looking over at this region over here, -6 and down, that only has one of these statements. It only has the x is less than or equal to, -5 so both aren’t included, so this region doesn’t count.

Between -6 and 5, they’re both represented so this is part of our intersection. And greater than 5, again we only have one element. So the intersection is where these two overlap so it’s really just this middle region. We’re dealing with from -6 to 5; both end points are included so we include both.

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