Extra Problems: R.1 through R.7

These are example problems, mostly from tests and quizzes that I have given in the past. Make sure you understand and can do all assigned textbook problems, activities, problems from handouts, and quizzes in addition to these problems.

1.) Write each of the following sets in interval notation.

1. {x| 7 < x ≤ 10}
2. {x | x > -2}

2.) True or False: Write true if the statement is always true for the indicated conditions. Write false otherwise.
(a. & b.) If x > 0 and y < 0:

1. _________________
2. __________________
3. If __________________

1. Assume x < 0,   |5x³|
2. |3x²|
3. Assume x > 3, |3 - x|

4.) Rewrite the following sentence as an inequality involving an absolute value.

The distance between -1 and x is at most 2.

5.) Evaluate x²y³ + xy² – y – x² where x = 2 and y = -1.   (I recommend that you do this one without a calculator.)

6.) Simplify the following completely. Write all your answers in exponential form. Do not have negative exponents in your answer. Cancel everything possible. No variable should appear more than once in your answer. Assume all variables are positive. Get rid of all parentheses.

7.) Rewrite the following expressions as one radical with everything possible pulled out. Rationalize all denominators. Assume all variables are positive.

8.) Simplify the following radicals assuming x and y could be any real numbers.

9.) Simplify the following expressions. Write your answer in regular polynomial form.

10.) Factor and simplify the following expressions completely.

1. 
2. 
3. 
4. (a + 2b) ^-1/3(a - b)^3/4 - (a + 2b)^2/3(a - b) ^-1/4

11.) Perform the indicated operations and simplify. Do not have complex fractions in your answer. Write your answer as a single fraction and reduce it completely. Rationalize the denominator where appropriate.

12.) Simplify the following expressions completely. Do not have negative exponents in your answer.

13.) Simplify . Then rewrite your answer using radical notation.

14.) Write 20,300 in scientific notation.

15.) Subtract 2x² – 3x + 2 from x² + 2x – 1. Simplify your answer.

16.) Multiply (2a + b) by (a² – 3b). Simplify your answer.

17.) Solve each of the following for x algebraically for full credit or graphically (where possible) for half credit.

1. 7x² – 1 = 0
2. (x + 1)³ = x³ + 1
3. 2x³ + 3x² – 2x – 3 = 0

18.) Fill in the blanks.

1. Put each of the following sets in order as indicated: ℂ, ℚ, ℝ, ℤ, and :ℕ
   ______ ⊂ _______ ⊂ ______ ⊂ ______ ⊂ ______
2. Fill in the blank with an example: ______ ∈ :ℝ\ℚ

19.) Name the mathematical property indicated:

1. k(x + y) = kx + ky
2. x + y = y + x
3. a(bc) = (ab)c

Last Update: August 18, 2010

Answers:

1. a.) (7, 10]     b.) (-2, ∞)
2. a.) false   b.) true   c.) true
3. a.) -5x³     b.) 3x²     c.) x - 3
4. 
5. 2²(-1)³ + 2(-1)² – ^-1 - 2² = -4 + 2 +1 – 4 = -5
6. a.) b.) c.)
7. a.) b.)
8. a.)     b.) |x + 3|     c.)
9. a.) 6x³ -11x² - 18x + 20     b.) 8x³ + 12x²y + 6xy² + y³     c.) -4xy - 3y²z
10. a.) (3 - x)(1 - yz)     b.) (4x - y)²     c.) 2(x - 2)(x² + 2x + 4)
    d.)
11. a.)     b.)     c.) -1     d.)     e.)
12. 1. 
    2. 
    3. 
    4. 
13. 
14. 2.3 × 10⁴
15. (x² + 2x –1) – (2x² – 3x +2) = -x² + 5x - 3
16. (2a + b)(a² – 3b) = 2a³ – 6ab + a²b – 3b²
17. 1. ±√(7)/7
    2. 0, -1
    3. -3/2, -1, 1
18. 1. ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
    2. √(2) (answers will vary)
19. 1. distributive property
    2. commutative property of addition
    3. associative property of multiplication