Exercises

Work through the following together. Be sure that each person in the group knows how to do the problem and can explain it to the others. Take turns explaining the problems. Don't be afraid to ask each other to repeat an explanation. Even if you can do the intermediate steps in your head, explain them carefully so that you could explain them at the blackboard. In your write up, write out the intermediate steps. Everyone in the group should be working on the problems and writing out solutions. However, one person should be a designated ``scribe'' for the version that you hand in.

1. First, let's think about some basic properties of absolute value.
   1. If |x| = 3, what does that tell us about x? If |x| = c, where c represents a real number, what does that tell us about x?
   2. If , what does that tell us about x?
2. The description of each of the following sets A involves inequalities. Rewrite the set as a union of intervals. (That is, write them in the form .)

   

3. The description of each of the following sets A involves inequalities. Rewrite the set as a union of intervals.

   

4. How do you find the values of x that satisfy a quadratic equation, ? What are the possibilities for solutions to the equation? What do the possibilities depend upon? What does this have to do with factoring the polynomial?
   1. Write A as a union of intervals, where
   2. Write A as a union of intervals, where
5. As a group, think of three examples of descriptions of sets like the ones we have been doing, but not identical to any of them. (So it is not enough just to change the numbers.) Write the sets as intervals, then exchange your examples with those of a neighboring group. Do those of the other group. We will put some of the solutions on the board at the end of class.