MA005: Calculus I

1. (a) $x = 2, 4$

(b) $x = –2, –1, 0, 1, 2, 3, 4, 5$

(c) $x = –2, –1, 1, 3$

3. (a) all $x$ (all real numbers)

(b) $x > 3 –2$

(c) all $x$

5. (a) $x = –2, –3, 3$

(b) no values of $x$

(c) $x ≥ 0$

7. (a) If $x ≠ 2$ and $x ≠ –3$, then $x^2 + x – 6 ≠ 0$. True.

(b) If an object does not have 3 sides, then it is not a triangle. True.

9. (a) If your car does not get at least 24 miles per gallon, then it is not tuned properly.

(b) If you can not have dessert, then you did not eat your vegetables.

11. (a) If you will not vote for me, then you do not love your country.

(b) If not only outlaws have guns, then guns are not outlawed. (poor English If someone legally has a gun, then guns are not illegal.

13. (a) Both $f(x)$ and $g(x)$ are not positive.

(b) $x$ is not positive. ( $x ≤ 0$ )

(c) 8 is not a prime number.

15. (a) For some numbers $a$ and $b, | a+b | ≠ | a | + | b |$.

(b) Some snake is not poisonous.

(c) Some dog can climb trees.

17. If $x$ is an integer, then $2x$ is an even integer. True.
Converse: If $2x$ is an even integer, then $x$ is an integer. True.
(It is not likely that these were the statements you thought of. There are lots of other examples).

19. (a) False. Put $a = 3$ and $b = 4$. Then $(a + b)^2 = (7)^2 = 49$, but $a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$.

(b) False. Put $a = –2$ and $b = –3$. Then $a > b$, but $a^2 = 4 < 9 = b^2$.

(c) True.

21. (a) True.

(b) False. Put $f(x) = x + 1$ and $g(x) = x + 2$. Then $f(x)g(x) = x^2 + 3x + 2$ is not a linear function.

(c) True.

23. (a) If $a$ and $b$ are prime numbers, then $a + b$ is prime. False: take $a = 3$ and $b = 5$.

(b) If $a$ and $b$ are prime numbers, then $a + b$ is not prime. False: take $a = 2$ and $b = 3$.

(c) If $x$ is a prime number, then $x$ is odd. False: take $x = 2$. (this is the only counterexample)

(d) If $x$ is a prime number, then $x$ is even. False: take $x = 3$ (or 5 or 7 or ...) 

25. (a) If $x$ is a solution of $x + 5 = 9$, then $x$ is odd. False: take $x = 4$.

(b) If a 3–sided polygon has equal sides, then it is a triangle. True. (We also have nonequilateral triangles .)

(c) If a person is a calculus student, then that person studies hard. False (unfortunately), but we won't mention names.

(d) If $x$ is a (real number) solution of $x^2 – 5x + 6 = 0$, then $x$ is even. False: take $x = 3$.