Download Math M01 and Math M03

M01 & M03 Problems
Simplify.
1a. ( −12x – 20y) – (25x – 20y)
b.
3
4
c+
7
2
=
1
2
c −
1
2
c. 3(4x – 2) – (5x – 8) = 8 – (2x + 3)
Solve for y.
2. −7x + 3y ≤ 21
Solve the compound inequality. Graph the solution set.
3. −8 ≤ 7x – 1 ≤ 13
4. The sum of three consecutive even integers is − 108. Find the numbers. (Specify what x is!)
5. A man has a collection of dimes and quarters with a total value of $3.50. If he has 7 more dimes than
quarters, how many of each does he have?
6. A woman has money in two accounts. One account pays 7% annual interest, whereas the other pays 9%
annual interest. If she has $600 more invested at 9% than she does at 7% and her total interest for a year is
$182, how much does she have in each account?
Complete the given ordered pairs and use the results to graph the solution set for the equation.
7. y =
1
2
x − 3
(−2, __ )
(0, __ )
( __ , −2)
Find the slope and the y-intercept for the following.
8. 3x – 4y = 8
9a. Find the equation of the line passing through the points (−3, 1) and (−2, 4).
b. What would be the slope of a line that is parallel to the line in part (a)?
c. What would be the slope of a line that is perpendicular to the line in part (a)?
Graph the inequality.
10. 2x + 3y ≤ 6
Solve by (a) graphing, (b) using the elimination method, (c) using the substitution method.
11. 5x − 2y = 10
x–y= −1
Simplify the following.
12a. (
c.
2 4 5 3
ab)
3
(5𝑥𝑥 4 𝑦𝑦4 )(10𝑥𝑥 3 𝑦𝑦3 )
(25𝑥𝑥𝑦𝑦 5 )(−2𝑥𝑥𝑦𝑦 7 )
b.
d.
(𝑑𝑑−4 )−8 (𝑑𝑑2 )−5
(𝑑𝑑3 )−4
40𝑥𝑥 10 𝑦𝑦 10
8𝑥𝑥 2 𝑦𝑦 5
+
10𝑥𝑥 8 𝑦𝑦 8
5𝑦𝑦 3
Simplify.
13. (4x
3
Multiply.
−
2
5
3
9
1
5
x2 + x – 1) – ( x3 + x2 – x + )
8
2
4
6
(7a4 – 8)(4a3 – 6)
14a.
b. (3x
– 5)2 – (2x + 3)2
Simplify the numerator, and then divide.
5m2 (6m−3)+ 6m3 (3m−1)
15a.
b.
3m
6v2 +5v+1
2v+3
Solve each equation.
16a. |5x – 3| + 4 = 3
b. |5x – 3| − 4 = 3
d. |3x + 5| − 8 < 5
c. 2|3x – 1| = 8
Factor the greatest common factor from the following.
2
2 2
2
2 3
5 3
4 4
17a. 5ab + 10a b + 15a b
b. 12x y – 72x y – 36x y
Factor by grouping.
2
18a. 20x + 4x + 25x + 5
3
2
b. 6x – 4x + 15x – 10
Factor the following trinomials completely.
2
2
19a. y + 3y – 18
2
b. 100p – 1,200p + 3,200
2
2
c. x – 13xb + 36b
d. x + 9x +
2
2
20a. 6x – 13x + 6
81
4
b. 15t – 79t – 34
Factor the following completely. Look first for the greatest common factor.
4
3
2
21a. 10x + 7x – 12x
3
Factor the following.
2
2
22a. 49y – 25z
2
b. 35y – 60y – 20y
b. h − 16
4
3
3
23a. 27 – x
b. 10r + 1,250
Solve the following equation.
24. 4y3 – 2y2 − 30y = 0
Simplify each side as much as possible, then solve the equation.
25. 3x(x – 3) = 2x(x – 4) + 6
Use factoring by grouping to solve the following equation.
26. x3 + 5x2 – 9x – 45 = 0
Solve the following word problem. Be sure to show the equation used. Define x!
27. The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by two consecutive
even integers. Find the lengths of the two legs.
Reduce the following rational expressions to lowest terms, if possible.
28a.
2m3 +4m2 −6m
b.
m2 −m 12
3a2 −8a+4
c.
9a3 −4a
x2 −6x+ax−6a
x2 −7x+ax−7a
Multiply or divide as indicated. Be sure to reduce all answers to lowest terms. (The numerator and denominator
should not have any factors in common.)
𝑐𝑐 2 − 5𝑐𝑐
29.
a.
30.
𝑚𝑚2
𝑐𝑐 2 + 7𝑐𝑐 +12
2𝑚𝑚 − 1
+ 𝑚𝑚 − 6
÷
−
𝑐𝑐 3 − 7𝑐𝑐 2 + 10𝑐𝑐
𝑚𝑚2
𝑐𝑐 2 + 9𝑐𝑐 + 18
b.
𝑑𝑑2 + 𝑑𝑑 − 42
4𝑑𝑑2 + 31𝑑𝑑 + 21
÷
4𝑑𝑑2 − 5𝑑𝑑 − 6
𝑑𝑑3 − 8
𝑚𝑚 + 2
+ 5𝑚𝑚 + 6
31. A solution contains 15 milliliters of HCl and 42 milliliters of water. If another solution is to have the same
concentration of HCl in water but is to contain 140 milliliters of water, how much HCl must it contain?
32.
1−
33.
4
1− 2
𝑦𝑦
2
2
𝑦𝑦
−
=
𝑦𝑦 2 − 9
3
8
𝑦𝑦2
5
𝑦𝑦 2 − 3𝑦𝑦
4
4
b. − √81
34a. √−125
c. − √−16
Simplify.
35a.
2
3
√54𝑥𝑥 3
b.
3
x √8𝑥𝑥 5
c.
3
125x
� 64y
d.
2√27a b
√9
Put each of the following radical expressions into simplified form. Assume all variables represent positive
numbers.
8√50
b.
5�27x y
7√75𝑎𝑎 𝑏𝑏
d.
3
36a.
c.
16√7
6
2
�7/9
Simplify the following.
37a.
1
2
1
√24 + 5 √150
b. 5√50 + 8√12 − √32 c. 9�24𝑥𝑥 3 𝑦𝑦 2 − 5x�54𝑥𝑥𝑦𝑦 2
d. 6�44𝑥𝑥 3 𝑦𝑦 3 − 8x�99𝑥𝑥𝑦𝑦 3 − 6y�176𝑥𝑥 3 𝑦𝑦
Use the properties of exponents to simplify the following expressions.
38a.
81/3
•
251/2
b.
(81y8)1/4
c.
(25a8b4)1/2
39.
√5 − √2
√5 + √2
40. √5𝑥𝑥 − 1 = 5
41. √𝑥𝑥 + 6
= x+4
Solve each of the following equations. Apply the square root property for equations. (Consider positive and
negative roots!)
2
42a. x + 8x + 16 = 25
b.
Solve by completing the square.
(a −
3 2
)
7
=
18
49
2
43. 5y – 10y = 4
Find the distance between the following points.
44a. (−3, − 8) and (−1, 6)
2
b. Find x so the distance between (−2, 3) & (x, 1) is 3.
2
45. Sketch the graph of x + y − 6x + 4y − 3 = 0 . Make sure to identify the center and radius.
46. Use the Quadratic Formula to solve the following:
(4x – 5)(x – 3) = 6
Divide the following complex numbers.
47.
–2+𝑖𝑖
5+6𝑖𝑖
Write the following radicals as complex numbers.
48. √−𝟐𝟐𝟐𝟐𝟐𝟐
Solve the following quadratic equation.
49.
1
1
1
2
x
+ 20 x + 4 = 0
5
Graph the following equation. Begin by completing the square on the first two terms.
2
50. x + 2x − 3