Staplers are not available at the Department.
Instructions :
(1) Read each question carefully, and prepare your solutions on a rough draft. Your final solution should be well-written, convincing, readable and elegant, even though the work it took to get there was anything but. Pictures are very useful, and often essential for solving a problem, but the picture itself is not a proof.
(2) Remember that your solution should be a convincing explanation of why your answer is correct . In questions 4 and 5, for example, the answer itself is worth very little of the mark.
(3) It must be possible to read what you have written out loud, as complete sentences. A proof should start with the hypothesis and conclude with the conclusion. Interesting steps (such as using a definition or result proven in class) should be justified.
(4) You do not need to justify routine algebraic manipulations unless you are asked for a
“proof from the axioms”.
(5) The bonus question is optional; you may earn more than 100% on this homework.
(6) Remember: although you may discuss the homework with friends, you must submit your own work. So for example, if someone explained the solution to you, you must write up your own solution without reference to any notes. To do otherwise is to cheat yourself — your goal is to succeed on your own!
Last name:
First name:
1

2
Student number:
The following table is just for the grader.
Question 1 2 3 4
Max score
Your score
4 4 6 4
5 6 (Bonus)
6 2
1. (4 points) Prove that the axioms imply that for every x ∈
R
, ( − 1) x = − x .
Recall that − 1 is defined as the additive inverse of 1 , and − x is the additive inverse of x . You may use any other results proven from the axioms in the notes, class or DGD; in particular, that the additive inverse of an element is unique.

3
2. (4 points) Decide if each of the following statements is true or false. If true, prove carefully (you may cite results proven in the notes, DGD, class); if false, give a concrete counterexample.
(a) For every
(b) For every a, b, c, d a, b ∈
∈
R
, if a < b and c < d , then ac < bd .
R
, if b < a < 0 then | a | < | b | .

4
3. (6 points)
(a) Prove that if | x + 3 | > 2 and | 2 x + 9 | < 3 then | x + 6 | < 1.
(Hint: your proof should start with
“Suppose x ∈
R is such that | x + 3 | > 2 and | 2 x + 9 | < 3 ” and should end with “Therefore
| x + 6 | < 1 ”.)
(b) Is it true that if | x + 6 | < 1 then | x + 3 | > 2 and | 2 x + 9 | < 3? Prove or disprove (i.e. give a counterexample).

5
4. (4 points) Using the definition of the absolute value, find all x ∈
R such that | − x 2 + 3 x − 1 | = 1.
Your solution should be a convincing explanation that you have indeed found all x satisfying this condition, in complete sentences.
Verify your answer by sketching the graph of y = |− x
2
+3 x − 1 | .

6
5. (6 points) Find all x ∈
R for which x − 3 x 2 − 3 x + 2
<
2 x −
2 − x
1
.
You solution should be a convincing argument that you have indeed found all such x , in complete sentences. See the sample solutions for some tips on organizing your solution.

7
6. (Bonus) Let a, r ∈
R be such that r > 0. Let x ∈
R
. Prove that if for every ε > 0 we have
| x − a | < r + ε then | x − a | ≤ r . Can you further conclude that | x − a | < r ? Explain.