Chapter 1: Worksheet 1 Jack K. Cohen Colorado School of Mines

Symbol Problems, Inequalities

Suggested Problems Section 1.1: 1, 5, 11, 15, 17, 19

1. (1.56) Express the area A of a square as a function of its perimeter P. Sketch a graph of this function—label your axes.

2. (1.58) Express the volume V of a sphere as a function of its surface area S. Sketch a graph of this function—label your axes.

3. If a > 0, correctly remove the absolute value signs from | a|. Same for | - a|.

4. If a < 0, correctly remove the absolute value signs from | a|. Same for | - a|.

5. True confession time. Problems like 1.19 drive me crazy! One day, the Voice said, ``Use the symbols, Luke''. Huh? Symbols? Oh yeah, symbols ...algebra ...do a general case! Our inequality looks like B≤| Dx + E|≤C. By dividing by a positive constant, we can simplify to b≤| x - a|≤c (convince yourself!). The ``interesting'' case is 0≤b≤c, so assume this.

   By doing the cases x≤a and x > a separately, show that the solution to the simplified inequality, b≤| x - a|≤c, ( 0≤b≤c) is the union of the closed intervals, [a - c, a - b] and [a + b, a + c].

6. Why are these cases not ``interesting''?
   1. b≤| x - a|≤c with b > c.
   2. b≤| x - a|≤c with b < 0.

7. Write down the solution of the closely related inequalities:
   1. b≤| x - a| < c
   2. b < | x - a|≤c
   3. b < | x - a|  < c

8. (1.19) Use the general inequality solution of the previous problems to check up on 2≤| 4 - 5x|≤4. Don't forget to put the problem into the right form by ``dividing by a positive constant''.

9. (1.17) What happens to the general inequality solution if c→ + ∞? Check up on | 1 - 3x| > 2. (Hint: explain why this is equivalent to 2 < | 1 - 3x| < + ∞.)

10. (1.15) What happens to the general inequality solution if b = 0? Check up on | 3 - 2x| < 5. Is the solution of 0 < | 3 - 2x| < 5 the same as for the given inequality?