IMA 101: Basic Mathematics
2/ 2010
Homework #5
Topics covered: solving polynomial and rational equalities and inequalities (Sections
1.5, 1.6, 1.10 Lial & Miller)
You do not need a calculator to solve these problems. You must show your work to receive
full credit, and should clearly indicate your answer. Although you are welcome to compare
your methods and answers with other students, you are highly encouraged to attempt the
problems on your own first.
20 questions, worth 100 points total
I. Polynomials
Practice [5 points each]
1. Factor the following and write the type of factorization you used.
2. Factor the following and write the type of factorization you used.
3. Factor the following and write the type of factorization you used.
4. Factor the following and write the type of factorization you used.
5. Factor the following. (hint: use Pascal’s triangle)
Application [5 points each]
(See EXAMPLE 1 on pages 3-4)
6. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
7. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
8. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
9. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
II. Rationals
Practice [5 points each]
10. Simplify the following completely. Make sure to indicate any constraints on the
variable.
Homework #5
Page 1 of 4
IMA 101: Basic Mathematics
2/ 2010
11. Simplify the following completely. Make sure to indicate any constraints.
12. Simplify the following completely. Make sure to indicate any constraints.
13. Perform the indicated operation and simplify completely. Make sure to indicate any
constraints on the variable.
14. Perform the indicated operation and simplify completely. Make sure to indicate any
constraints on the variable.
15. Perform the indicated operation and simplify completely.
16. Perform the indicated operation and simplify completely.
17. Simplify the following complex fraction completely.
Application [5 points each]
(See EXAMPLE 2 on page 4)
18. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
19. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
20. Solve the following inequality. Indicate your answer on the number line and in
interval notation. (hint: your answer will be similar to but not the same as in the
example)
BONUS
Factor the following (hint: use Pascal’s triangle)
Homework #5
Page 2 of 4
IMA 101: Basic Mathematics
2/ 2010
EXAMPLE 1
Q. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
[first subtract from both sides of the inequality to get 0 on the right side]
[now factor the left side, note that -5*1 = -5, and -5+1 = -4]
[now illustrate on the number line where (z-5) and (z+1) are positive and negative]
(z + 1)
-
-
-
-
-
- -
-
-
-
- 0
+ + + + + + + + + + + + + +
+
(z – 5)
-
-
-
-
-
- -
-
-
-
-
-
+
-
- -
-
-
0
+ + + + + + + +
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[Recall that a negative multiplied by a negative is positive, a negative multiplied by a
positive is negative, and a positive multiplied by a positive is positive.]
[now we can add to our drawing the sign of the product of (z+1) and (z – 5)]
(z + 1)(z – 5)
+ + + + + + + + + + 0
-
- -
-
-
0
+ + + + + + + +
+
(z + 1)
-
-
-
-
-
- -
-
-
-
- 0
+ + + + + + + + + + + + + +
+
(z – 5)
-
-
-
-
-
- -
-
-
-
-
-
+
-
- -
-
-
0
+ + + + + + + +
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[since we are looking for when
, we simply consider the intervals where
we wrote + signs. Note that since we are looking for “> 0”, or strictly greater than 0, we do
not include the points where we wrote “0” ]
[your final answer should be indicated on the number line and in interval notation]
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Homework #5
Page 3 of 4
IMA 101: Basic Mathematics
2/ 2010
What I expect to see on your homework:
Q. Solve the following inequality. Indicate your answer on the number line and in
interval notation.
a.
We are looking for values of r that will make this positive or zero. Since when r=5,
the denominator is zero, we must remember that r ≠ 5 (as indicated by the X below).
(r)/(r – 5)
+ + + + + + + + + + + 0
- -
-
-
X
+ + + + + + + +
+
r
-
-
-
-
-
- -
-
-
-
-
- 0
+ + + + + + + + + + + + +
+
(r – 5)
-
-
-
-
-
- -
-
-
-
-
-
- -
+
-
-
-
0
+ + + + + + + +
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
EXAMPLE 2
Q.


(z + 1)(z – 5)
+ + + + + + + + + + 0
-
- -
-
-
0
+ + + + + + + +
+
(z + 1)
-
-
-
-
-
- -
-
-
-
- 0
+ + + + + + + + + + + + + +
+
(z – 5)
-
-
-
-
-
- -
-
-
-
-
-
+
-
- -
-
-
0
+ + + + + + + +
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Homework #5
Page 4 of 4