Download Unit 1: Expressions, Simple equations, Inequalities and Absolute

Unit 1: Expressions, Simple equations, Inequalities and Absolute Value (1)
1.1 Simplifying and Evaluating Expressions
Simplify
1. a. 11 – 3 + 5 – 2
2. a. 12 – 5 – 2 + 3
3. a. 3 · 8 + 4 · 5
4. a. 4² - 6 ÷ 2 + 3
5. a. 6 · 4 + 5 · 2
b. 11 – (3 + 5) – 2
b. 12 – (5 – 2) + 3
b. 3 · (8 + 4) · 5
b. (4² - 6) ÷ 2 + 3
b. 6 · (4 + 5) · 2
6. 6 – [7 – (5 – 2)]
7. 14 – 2[9 – 2(5 – 3)]
9.
32
5  (3  1)
10.
c. 11 – (3 + 5 – 2)
c. 12 – (5 – 2 + 3)
c. 3 · (8 + 4 · 5)
c. (4² - 6) ÷ (2 + 3)
c. 6 · (4 + 5 · 2)
23  1
8. 3
2 1
2 2 (32  4 2 )
11.
10 2
1 1  72
3 52
12. 64 ÷ 4² + 3(3² - 1)
13. 2² · 3² - (5² - 4²)
14. [3³ - (2³ + 2²)] ÷ 5
1
[2(3 + 4) - 3²]
10
1 1  92
18.
2 52
16. 3 · 2³ - (7² - 5²)
17. [4(5 – 2) + 2³] ÷ 2
15.
19.
33
4  (4  1)
20.
43  6
43  6
Evaluate each expression if a = 6, b = - 2, x = 3, y = 2 and z = 5
21. 2 x 2  x  2
22. 3 y 2  y  5
23. ( yz  x) 3
24. ( xz  zy ) 3
25. 2 a + b
26. a - 3 b
27. a  b
28. a 2  b 2
xz x y
29.

y
2z
 xyz 

30. 
x yz
z2
y2
33.

x y zx
 z2  y2  x2
34. 
xy

2
4
z 2  (x2  y 2 )
3y 2 z
32.
4z 3
x2  y2
z2  y2
35.
xz  2 y( z  x)
36.
4 xyz
z2  x2
31.



5
2
Simplify by combining similar terms.
37. 8c + 2(c + 3)
40. 2(1 – y) + 4(3 – y)
43. (-2r + s + 5) + 2(r – 3s + 2)
38. 5(d + 2) – 3d
41. 7p – 4q + 3q – 10p
44. (6x – 5y + 4) + 2(-2x + 3y – 2)
39. 7(x + 2) + 4(x – 4)
42. 6m – 4n + (– 7)m – (- 5)n
1.2 Solving Equations in One Variable
Solve. Check your work when there is a single solution.
1. 3x – 4 = 5
2
t–8=0
3
2. 4z + 11 = 3
3.
5. 18 + 2r = 5r
6. 24 – 2y = 6y
7. 3(t – 1) = - (t - 5)
8. 2(x – 3) = x + 3
9. 2k – 1 = k – 5 + 3k
10. 3(2z – 1) = 6z + 5
11. – (5 – x) = x + 3
12.
4. 15 -
1
d  1
8
3
15
(u  2) 
2
2
3
1
9
(2r  3)  r 
16.
10
5
10
13.
14.
1
( s  2)  s  4
3
1
15. ( x  5)  x  5
5
2
18. ( x  2)  x  4
5
8
72
(2t  1) 
5
5
17. 3(x – 2) – x = 2(2x + 1)
19. 2z – (1 – z) = 11 – z
20. 3(1 – t) + 5 = 3(1 + t) – 7
22. 3(5z – 1) + 5(3z + 2) = 7
23.
25. 5(x – 2) – 3 = - (x + 1)
26. 2x – 1 = 2(x + 4)
27. 4a + 1 = a – 5 – 3a
29. 3(x – 2) = 5(x – 2)
30.
28.
6
48
(u  2) 
5
5
31. 11x – 3(4x – 2) = 2(8 + 2x)
34.
21. 2(5t – 3) – t = 3(3t – 2)
6 x  2( x  4)
8
3
24.
1
( x  3)  x  5
5
5k  3(k  2)
 6
33.
4
32. 2(x + 2) = 2x + 4
2
3
18
(2r  3)  r 
5
5
5
3 y  2( y  1)
 1
6
35. 4x – (x – 4) = 4(x – 4)
36. 5z – (6 – z) = 4 (1 – z)
37. Determine whether the following statement is true or false. If true, tell why. If false, give a counterexample. If
a statement a ≠ b is true, then a + c ≠ a + b is true for any real number c.
Mixed Review Exercises
Simplify
1. 7 – (3 – 4)
5. (- 8)(
1
)(- 3)(- 1)
2
9. (- 4a)(5b) + (- 3a)(- 8b)
2. (4 – 9)²
6.
 3  11
1 3
10. 2(c – 3d) + 5(2d – c)
3. – 2(- 5 + 8)
4.  6  2
7. 3x – 2(x + 4)
8.
11. – 3 +
1
(6 – 10x) + 5x
2
1 m
1
12. ( x) 2 ( y) 3
1.3 Solving Inequalities in One Variable
Solve each inequality and graph each solution set that is not empty. (#1-26)
1. x – 7 > - 5
2. y + 4 < 3
3. 5(x – 7) + 2(1 – x) > 3(x – 11)
4. 2t < 6
5. 3u > - 6
6. 7y – 2(y – 4) > 6 – (2 – y)
7. – 5x < 10
8. – 12 > - 4y
9. 4s + 3(2 – 3s) < 5(2 – s)
10. -
t 3

2 2
11. -
3
k  6
4
12. 4(2 – x) – 3(1 + x) < 5(1 – x)
13. 3s – 1 > - 4
14. 2r + 5 < - 1
15. k – 3(2 – 4k) < 7 – (8k – 9 + k)
16. y < 7y – 24
17. 3t > 6t + 12
18.
19. 2 – h < 4 + h
20. 1 + 2x < 2(x – 1)
21. 4(y + 2) – 9y > y – 3(2y + 1) – 1
22. 5(2u + 3) > 2(u – 3) + u
23. 3(x – 2) – 2 < x – 5
24. 4[5x – (3x – 7)] < 2(4x – 5)
25. 5z + 11 < 1
26. 7c – 9 < 4(2c – 3)
2
t  (2  3t )  5t  2(1  t )
3
Tell whether each statement is true for all real numbers. If you think it is not, give a numerical example
to support your answer. (#27-36)
27. If a < b, then a – c < b – c
28. If a < b, then a – b < 0
29. If a < b, then a 2  b 2
30. If a < b, then a 3  b 3
31. If a > b, then b < a
32. If a < 0, then a² > 0
33. If a < b and c < d, then a + c < b + d
34. If a < b and c < d, then a - c < b - d
35. If a > b and c < 0, then ac < bc
36. If ac = bc and c ≠ 0, then a = b
37. The inequality – 2x < 4x + 12 gives a certain chemicals safe temperature, x, in degrees Celsius. At
what temperature is the chemical safe?
38. Kevin says the solution to y – 9 > - 18 is y < - 9. What error has he made?
Match each inequality with the graph of its solution set.
39. – 6x > 3
40. 4 – x > 2
41. 1 – x < x
42. 2(x – 1) < 2x – 3
43. 2(x + 1) + 1 > 0
44. 2(x – 1) > 2x - 3
1.4 Solving Combined Inequalities
Solve and graph each solution set that is not empty (#1 – 22)
1. 2t + 7 ≥ 13 or 5t – 4 < 6
2. 2t + 7 ≤ 13 and 5t – 4 > 6
d
4. – 3 < 2 ≤-1
3
6. x – 7 < 3x – 5 < x + 11
3. – 5 < 1 – 2k < 3
5. 7q – 1 > q + 11 or – 11q > - 33
3
3
7. - m ≥ m – 1 or - m < m + 1
4
4
9. – 3 ≤ - 2(t – 3) < 6
8. 1 ≤ 3x + 4 ≤ 13
10. – 5 < 2(2 – s) + 1 ≤ 9
11. 2x + 3 > 1 or 5x – 9 ≤ 6
15. 5n – 1 > 0 and 4n + 2 < 0
12. 2x + 3 < 1 and 5x – 9 ≤ - 6
n
14. 3 ≥ 1 - > - 2
2
16. 3y + 5 ≥ 2y + 1 > y – 1
17. 3z + 7 ≤ 4z and 3z + 7 > - 4z
t
t
19.  2  t  3 and t  3   4
4
2
1
21. m  1 and 5 – m > 1
2
18. p + 2 < - 1 or – 4p ≤ - 8
r 3
r 6
 r  1 or
r4
20.
6
3
2x  3
7
22.  9 
4
13. – 6 ≤ 2 – 3m ≤ 7
23. Explain why it would be incorrect to write – 2 < x < - 5?
24. Which pair of numbers belongs to the solution set of 3 < 5x + 3 ≤ 8?
A) 0 and 8
B)
1
and 1
2
C)
3
8
and
5
5
D) 0 and 1
25. Which pair of numbers in interval notation gives 0 ≤ 2x – 6 ≤ 4?
A) [- 3, 5]
B) [5, 3]
C) [3, 5)
D) [3, 5]
Match each graph with one of the open sentences in a – h.
a. – 2 ≤ x ≤ 3
e. – 2 < x < 3
b. x < 0 or x > 0
f. x ≥ - 2 or x ≤ 3
c. x > 3 or x ≤ - 2
g. – 2 ≤ x < 3
d. x < - 2 or x ≥ 3
h. x ≤ - 2 or x > 3
1.5 Solving Absolute Value sentences
Solve each open sentence and graph the solution set.
1. │2t + 5│< 3
2. │3x + 2│> 4
3. │2u - 5│= 0
4. 8 =│5y + 2│
5. │3 - x│≥ 2
6.│2 - p│≤ 2
7. 0 ≤│4u - 7│
8. │3r - 12│> 0
9. │t - 2│≤ 2
10. 1 >│2 - 8n│
11. │x + 5│- 3 = 1
12. │2w│+ 4 = 8
13. │u│ + 4 = 5
14. 15 =│3 – 6x│
15. │2x│- 3 = 5
16. 12 -│4w│ = 2
17. │w│+ 4 ≥ 6
18. │t + 4│= 0
19. │5 - 2n│= 3
20. │6y - 6│> 0
21. │2t - 3│+ 2 = 5
22. │2u - 1│+ 3 ≤ 6
23. -│3k + 1│ + 4 < 2
24. 7 - 3│4d - 7│≥ 4
25. 6 + 5│2r - 3│≥ 4
26. 4│3t - 5│ + 8 > 5
27. 6│2t - 5│-9 ≥ 15
28. 2│2x – 7│ + 11 = 25
29. │m + 5│ + 9 ≤ 16
30. │t - 7│ + 3 ≥ 4
31. │1 – 2x│+ 6 = 9
Solve and give the letter to the correct answer for each problem below
32. If y is an integer, what is the solution set for │y + 5│= 4?
A) {-1, - 9}
B) {-1, 9}
C) {1, -9}
D) {1, 9}
33. What is the complete solution to the equation │15 - x│= 21?
A) x = - 6, x = 36
B) x = - 6, x = - 36
C) x = 6, x = - 36
D) x = 6, x = 36
34. What are the possible values of h in │4 - h│≥ 5?
A) h ≤ - 1 or h ≥ 9
B) h ≤ - 1
C) h ≤ - 9 or h ≥ 1
D) – 1 ≤ h ≤ 9
35. What is the solution to the inequality │2y + 9│< 13?
A) – 2 < y < 11
B) y > 2
C) – 11 < y < 2
D) y < - 11 or y > 2
36. Describe the graph of the solution set of │k + 3│< 2?
A) points at least 2 units from 3
B) points less than 2 units from - 3
C) points more than 3 units from 2
D) points at most 2 units from – 3
37. Which of the following values below give the solution to the equation │2x + 6│= 14?
A) x = - 10, x = 4
B) x = - 4, x = 10
C) x = - 31, x = 25
D) x = - 25, x = 31
*Reminder For every journal entry:
1. Date each journal entry with the date it was assigned.
2. Copy the question assigned.
3. Answer the journal questions in complete sentences using your own words and until you feel
you have completely answered the questions. If the journal requires you to solve a problem do so
and explain how to solve the problem if required.
4. MAKE SURE YOU COMPLETE YOUR JOURNALS BEFORE THEY ARE DUE!
Unit 1 Journal topics
1. a. Solve and describe the steps to solving the equation:
1x – 2(3x + 1) = 7(4 – 5x).
2. Observe the inequalities(*Notice how they are alike, different):
9 – 2(x + 6) > 25 and 9 – 2(x + 6) ≥ 25
* List the similarities and differences between the inequalities.
* Solve and describe how the solutions of the inequalities are alike? different?
3. How is solving an inequality different from solving an equation (consider all
cases)? How are their solutions different?
4. Do you think algebra is an important subject for you to study?
Why or why not?
Explain fully.
5. Observe the absolute value equations(*Notice how they alike, different):
a. │2x + 1│= 5
b. │2x + 1│< 5
c. │2x + 1│> 5
*Solve each open sentence and graph its solution set.
*Was your approach to solving each open sentence similar or different, explain.
*How are the solutions to these absolute value problems similar? different? Explain.