Download Intermediate Algebra Review for Test 4

Intermediate Algebra
Review for Test 4
Determine the domain of the rational function.
7
x-7
1) a) f(x) =
b) h(x) =
x+8
2
2x + 7x - 9
c) g(x) =
9x - 6
x2 - 49
d) p(x) =
1 - 8x
x3 - 14x2 - 15x
Simplify the rational expression.
2) a)
c)
Find R(x) = f(x) · g(x) and state its domain.
x2 + 5x - 6
x2 - 4x - 21
6) f(x) =
, g(x) =
x-7
2x2 + 11x - 6
2x + 2
2
10x + 16x + 6
b)
a 2 + 5a + 6
a+2
18x - 6
15x3 - 5x2
d)
x2 + 5x + 6
x2 + 6x + 9
Divide the rational expression. Express the quotient as a
rational expression in lowest terms.
5a 2
x2 - 9
9x - 6
x2 - 49
d) p(x) =
1 - 8x
x3 - 14x2 - 15x
Simplify the rational expression.
4) a)
c)
Find R(x) =
b)
a 2 + 5a + 6
a+2
18x - 6
15x3 - 5x2
d)
x2 + 5x + 6
x2 + 6x + 9
b)
z 2 + 16z + 64 z 2 + 3z
·
z+8
z2 - 9
c)
x2 - 13x + 30 x2 - 17x + 66
·
x2 - 13x + 42 x2 - 15x + 50
y2 - 25 y2 + 9y + 18
·
y2 - 9 y2 - 8y + 15
e)
4m 2 + 8mn + 4n 2
n-m
·
2
2
5m
+ 5n
m -n
f)
z3 - 125
2z2 + 5z - 33
·
z 2 - 8z + 15 9z 2 + 45z + 225
d)
7y - 70
49
3-x
15xy
y2 + 5y - 6
y2 + 9y + 18
y2 - 1
y2 + 7y + 12
f(x)
and state its domain.
h(x)
x2 - 5x + 4
x-1
, h(x) =
x+3
2
5x + 13x - 6
Perform the indicated operation and simplify the result.
5x
2
9) a)
4x + 9 4x + 9
Multiply the rational expression. Express the product as a
rational expression in lowest terms.
96x
8x - 24
5) a)
·
2
x -9
12x2
d)
(y - 10)2
7
8) f(x) =
2x + 2
2
10x + 16x + 6
3y
b)
4a
3b2
c)
Determine the domain of the rational function.
7
x-7
3) a) f(x) =
b) h(x) =
x+8
2
2x + 7x - 9
c) g(x) =
b3
7) a)
b)
8x
3
+
x-5 5-x
c)
x2 - 8x - 37 x2 - 3x - 3
x-8
8-x
Add or subtract, as indicated, and simplify the result.
7
2
10)
+
27x 3x2
11)
y- 4 y- 4
y+ 3 y+ 1
12)
1
3
(x - 4)(x - 1) (x - 1)(x - 2)
13)
x
x2 - 16
-
4
x2 + 5x + 4
Page 2
Perform the indicated operations and simplify the result.
5x
6
10
14)
+
x + 1 x - 1 x2 - 1
Simplify the complex rational expression.
15) a)
2
+2
a
2
-2
a
1b)
5
x
25
xx
c)
36t2 - 49s2
st
6 7
s t
Solve the equation.
6
4
16)
=
p-8 p+8
1 30
=
y y2
19)
5
8
10
=
x + 2 x - 2 x2 - 4
20)
x-5
24
=
x
x+5
Evaluate the square root.
27) a) 64
b) - 625
1
c)
49
Tell if the square root is rational, irrational, or not a real
number. If the square root is rational, find the exact value;
if the square root is irrational, write the approximative
value rounded to 2 decimal places.
28) a) 169
b) -36
c) 43
1
1
b- 3
21) +
=
b b- 4 b- 4
Solve the problem.
22) a) For the function f(x) = 2x +
b) For the function f(x) =
Solve the motion problem.
25) A plane flies 600 miles west into the wind and
makes the return trip (against the wind)
following the same flight path. The effect of
the jet stream on the plane is 15 miles per hour.
The round trip takes 9 hours. What is the
speed of the plane in still air?
26) A boat moves 7 kilometers upstream in the
same amount of time it moves 20 kilometers
downstream. If the rate of the current is 5
kilometers per hour, find the rate of the boat in
still water. (Write your answer as a fraction or
a mixed number.)
6
17) m =5
m
18) 1 +
24) One pump can drain a pool in 11 hours. When
a second pump is also used, the pool only
takes 5 hours to drain. How long would it take
the second pump to drain the pool if it were
the only pump in use? (Write your answer as a
fraction or a mixed number.)
4
, solve f(x) = 9.
x
x+3
, solve f(x) = 2.
x-1
Solve the work problem.
23) A painter can finish painting a house in 7
hours. Her assistant takes 9 hours to finish the
same job. How long would it take them to
complete the job if they were working
together? (Write your answer as a fraction or a
mixed number.)
Simplify the square root.
29) a) 732
b)
c)
g2
(2x - 7)2
Evaluate the expression, if possible.
30) a) 815/4
b) (-1000)2/3
c) 36-3/2
Rewrite the expression with a positive rational exponent.
Simplify, if possible.
5
7
31) a) y
b)
(x + y )3
Page 3
32) a) 121/5 · 123/5
y3/4
b)
y1/4
c) x1/5 · x3/4
33) (25x2 y3/2)1/2
Simplify the radical expression. Assume that all variables
represent positive real numbers.
9
34) a) w10
b)
x54
Use the product rule to multiply. Assume all variables
represent positive real numbers.
35) a) 12 · 3
3
3
b) 2 · 16
3
3
c) 2 6 · 3 5
d) x + 4 · x - 4
5
5
e) 7x4 · 16
Use the product rule to simplify the expression. Assume
all variables represent positive real numbers.
3
36) a) 63
b)
500
2
c) 175x
d) 245k7 q8
Use the quotient rule to divide and simplify.
3
8
7x
37) a)
b)
49
3
2
Add or subtract. Assume all variables represent positive
real numbers.
38) a) 15 2 + 12 2
b) 80 + 320
4
4
c) 11 x7 - 3x x3
Multiply, and then simplify if possible. Assume all
variables represent positive real numbers.
39) a) 7( 5 + 3)
b) ( 6 + 5)( 6 - 5)
c) ( 5 + 2)2
Rationalize the denominator. Assume that all variables
represent positive real numbers.
6
18
81
40) a)
b)
c) 4
x
7
49x3
Rationalize the denominator and simplify. Assume that all
variables represent positive real numbers.
3
2
41) a)
b)
7- 6
5+ 7
Evaluate the radical function at the indicated value.
42) f(x) = 2x + 3
f(11)
Find the domain of the given function.
43) a) f(x) = 3x - 12
b) f(x) = 6 - 5x
Solve the equation.
44) a) x + 1 = 4
b) x - 3 = 4
c) 18y - 27 = y + 3
Solve the linear inequality. Express the solution using
interval notation. Graph the solution set.
45) a) x + 5 < 4
b) 5x - 2 > 4x + 4
6x + 6
c)
< -27
2
d)
3
1
x > (2x - 3)
4
5
Find the following for the given equation.
46) -2x + 5y = 10
a) the x-intercept
b) the y-intercept
c) the slope
d) Graph the equation.
Find an equation of the line containing the given pair of
points. Write your final answer in slope-intercept form.
47) (-3, -1) and (-1, -9)
48) (5, -4) and (10, -7)
Solve the absolute value equation.
49) |2x + 5| = 7
Page 4
50) |5x + 9| + 8 = 12
a) Find the constant of proportionality k, b) write the
equation of variation, then c) find the quantity indicated.
51) Suppose that y is varies directly with x. When
x = 18, then y = 288. Find y when x = 14.
Solve.
52) When the temperature stays the same, the
volume of a gas is inversely proportional to the
pressure of the gas. If the volume of a balloon
is 280 cubic inches of gas at a pressure of 14
pounds per square inch, find the volume when
the pressure is 56 pounds per square inch.
Solve the problem.
53) A certain aircraft can fly 1152 miles with the
wind in 4 hours and travel the same distance
against the wind in 8 hours.
a) What is the airspeed of the plane?
b) What is the speed of the wind?
54) Inclusive of a 6.7% sales tax, a diamond ring
sold for $2027.30. Find the price of the ring
before the tax was added. (Round to the
nearest cent, if necessary.)
55) When Milo got promoted at work, he received
a 20% pay raise. He now earns $45,600 per
year. What was his annual salary before his
raise?
56) After a 13% price reduction, a boat sold for $
23,490. What was the boat's price before the
reduction? (Round to the nearest cent, if
necessary.)
Answer Key
Testname: IAREVIEW4F12
1) a) {x|x ≠ -8}
9
b) {x x ≠ - , 1}
2
c) {x|x ≠ 7, -7}
d) {x|x ≠ 15, -1, 0}
1
2) a)
b) a + 3
5x + 3
c)
6
5x2
d)
d) {x|x ≠ 15, -1, 0}
1
4) a)
b) a + 3
5x + 3
5) a)
c)
5x2
6)
x+2
d)
x+3
64
x(x + 3)
b)
(x - 3)(x - 11)
(x - 7)(x - 5)
d)
(y + 5)(y + 6)
(y - 3)2
f)
2z + 11
9
e) -
4
5
z(z + 8)
z-3
(x - 1)(x + 3)
1
; {x|x ≠ 7, -6, }
2x - 1
2
7) a)
15a
4b
b) -5x(x + 3)
c) y - 10
y+4
d)
y+1
8) (x - 4)(5x - 2); {x|x ≠ 1, -3,
9) a)
5x - 2
4x + 9
8x - 3
b)
x-5
c) 2x + 5
7x + 18
10)
27x2
11)
-2(y - 4)
(y + 3)(y + 1)
13)
x2 - 3x + 16
(x - 4)(x + 4)(x + 1)
14)
5x - 4
x-1
15) a)
1+a
1-a
b)
1
x+5
c) 7s + 6t
16) - 40
17) {-1, 6}
18) {-6, 5}
19) {-12}
20) {25, -1}
21) {1}
1
22) a) x = or x = 4,
2
c) {x|x ≠ 7, -7}
c)
-2(x - 5)
(x - 4)(x - 1)(x - 2)
x+2
x+3
3) a) {x|x ≠ -8}
9
b) {x x ≠ - , 1}
2
6
12)
2
}
5
23)
63
15
or 3
hours
16
16
24)
55
1
or 9 hours
6
6
b) x = 5
25) 135 miles per hour
135
5
26)
or 10
kilometers per hour
13
13
27) a) 8
b) -25 c)
1
7
28) a) rational, 13
b) not a real number
c) irrational, 6.56
29) a) 73 b) |g|
c) |2x - 7|
30) a) 243
b) 100
1
c)
216
31) a) y1/7
b) (x + y)3/5
32) a) 124/5
b) y1/2
c) x19/20
33) 5xy3/4
34) a) w5
b) x6
Answer Key
Testname: IAREVIEW4F12
35) a) 6
b) 2
c) 6
45) a) (-∞, -1)
3
4
-8
3
30
x2 - 16
d)
5
e) 112x4
36) a) 3 7
3
b) 5 4
c) 5x 7
d) 7k3 q4 5k
2 2
37) a)
7
3
b) 2 x
38) a) 27 2
b) 12 5
4
c) 8x x3
39) a) 35 + 21
b) -19
c) 7 + 2 10
6 7
40) a)
7
b)
c)
41) a)
3 2x
x
3
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
21 + 3 6
43
b) 7 - 5
42) 5
43) a) [4, ∞)
6
b) (-∞, ]
5
44) a) {15}
b) {19}
c) {6}
5
6
1
2
3
4
5
6
7
8
9
10 11 12 13
b) (6, ∞)
-1
0
c) (-∞, -10)
-14
d)
-13
-12
-11
-10
-9
-8
-7
-6
12
,∞
7
-
-6
-5
-4
-3
-2
-1
0
46) a) x-intercept: (-5, 0)
b) y-intercept: (0, 2)
2
c) slope:
5
y
10
5
-10
-5
5
-5
4
49x
7x
4
-10
d)
47) y = -4x - 13
3
48) y = - x - 1
5
49) 1, - 6
50) - 1, -
13
5
51) a) k = 16
b) y = 16x
c) 224
52) 70 cubic inches
53) a) 216 mph
b) 72 mph
54) $1900
55) $38,000
56) $27,000
10
x
1
2
3