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Math 100 Half Chapter 8 and Chapter 9 Test Review Selected Solutions Perform the indicated operation and simplify the results.
1.
�
32x
+ 52x − 4
! 3.
�
2x + 5x2 − 2x
− 4x − 2
!
�
factor the denominators3
2x + 5
2(x − 2)
The LCD is 2x(x − 2)
multiply 32x
by (x − 2)(x − 2)
multiply 52(x − 2)
by xx
(x − 2)(x − 2)
•3
2x +xx
•5
2(x − 2)
3x − 62x(x − 2)
+5x
2x(x − 2)
combine the numerators over the LCD3x − 6 + 5x2x(x − 2)
simplify the numerator8x − 6
2x(x − 2)
factor the numerator2(4x − 3)2x(x − 2)
reduce(4x − 3)x(x − 2)
!
�
factor the denominators2x + 5x(x − 2)
− 4(x − 2)
The LCD is x(x − 2)
multiply 2x + 5x(x − 2)
by 11
multiply − 4(x − 2)
by xx
11
•2x + 5x(x − 2)
− xx
•4
(x − 2)
2x + 5x(x − 2)
− 4xx(x − 2)
combine the numerators over the LCD2x + 5 − 4xx(x − 2)
simplify the numerator−2x + 5
2x(x − 2)
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
5.
�
3x2 + 6x + 5
+ 2x2 + 3x + 2
!
�
factor the denominators3
(x + 5)(x + 1) + 2
(x + 1)(x + 2)
The LCD is (x +5)(x +1)(x + 2)
multiply 3(x + 5)(x + 1)
by (x + 2)(x + 2)
multiply 2(x + 1)(x + 2)
by (x + 5)(x + 5)
(x + 2)(x + 2)
•3
(x + 5)(x + 1) +
(x + 5)(x + 5)
•2
(x + 1)(x + 2)
3x + 6(x +5)(x +1)(x + 2)
+2x + 10
(x +5)(x +1)(x + 2)
combine the numerators over the LCD3x + 6 + 2x + 10
(x +5)(x +1)(x + 2)
simplify the numerator5x + 16
(x +5)(x +1)(x + 2)
!
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
6.
�
2x2 + 3x −18
− 3x − 3
watch the negative sign
�
factor the denominators2
(x + 6)(x − 3) − 3
(x − 3)
The LCD is (x +6) (x − 3)
multiply 2(x + 6)(x − 3)
by 11
multiply 3(x − 3)
by (x + 6)(x + 6)
11
•2
(x + 6)(x − 3) − (x + 6)
(x + 6)•
3(x − 3)
2(x + 6)(x − 3)
− (3x + 18)(x + 6)(x − 3)
combine the numerators over the LCD2 − (3x + 18)(x + 6)(x − 3)
distribute the negitive2 − (3x + 18)(x + 6)(x − 3)
2 − 3x −18(x + 6)(x − 3)
simplify the numerator−3x −16
(x + 6)(x − 3)
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
7.
�
x + 26x − 3
− x −18x − 4
watch the negative
�
factor the denominatorsx + 2
3(2x −1) − x −1
4(2x −1)
The LCD is 12 (2x −1)
multiply (x + 2)3(2x −1)
by 44
multiply (x −1)4(2x −1)
by 33
44
•(x + 2)
3(2x −1) − 3
3•
(x −1)4(2x −1)
4x + 812 (2x −1))
− (3x − 3)13 (2x −1)
combine the numerators over the LCD4x + 8 − (3x − 3)
12 (2x −1)
distribute the negitive4x + 8 − 3x + 3
12 (2x −1)
simplify the numeratorx + 11
8 (2x −1)
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
Solve for x.
10.
�
x − 33x
+ 12
= 32x
! 11.
�
4 − 8xx + 1
= 8x + 1
! !
�
the LCD is 6xmultiply each termby 6x and cancel
6x1
•(x − 3)
3x + 6x
1•
12
= 32x
•6x1
/ 6 2 / x 1
•(x − 3)
/ 3 / x + / 6 3x
1•
1/ 2
= / 6 3 x 1
•3
/ 2 x
2(x − 3) + 3x •1 = 3• 3
2x − 6 + 3x = 9
5x − 6 = 9
5x = 15
x = 3
�
the LCD is (x +1)multiply each termby (x +1) and cancel
(x +1)1
•41
− (x +1)1
•8xx + 1
= (x +1)1
•8x + 1
(x +1)1
•41
− (x +1)1
•8xx + 1
= (x +1)1
•8x + 1
4(x + 1) −1• 8x = 8
4x + 4 − 8x = 8
−4x + 4 = 8
−4x = 4
x = −1
when you plug −1 into theoriginal equation you geta zero in the denominator
No Solution (NS)
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
Chapter 9 Simplify.
13.
�
6 48 ! ! 15.
�
−5 40x 6y 3
! !
�
factor 48
6 16 • 3
16 = 4the comes out of the radical sign as a 4
6 • 4 • 3
= 24 • 3
!
�
−5 40 = −5 4 • 5 = −10 5
x6 = x2 • x2 • x2 = x3
y3 = y2 • y = y y
−5 40x6y3 = −10x3y y
!
!16.
�
24 − 2 54 + 3 18 ! 19.
�
15x3y⎛ ⎝
⎞ ⎠ 20x2y⎛ ⎝
⎞ ⎠
�
24 = 4 • 6 = 2 6
−2 54 = −2 9• 6 = −6 9• 6
+3 18 = +3 2• 3• 3 = +9 2
24 − 2 54 + 3 18
= 2 6 − 6 6 + 9 2
combineliketerms
= −4 6 + 9 2
!
�
15• 20• x3 • x2 • y • y
3• 5• 4 • 5• x5 • y2
2• 5• x2 • y 3• x
10x2y 3x
!
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
21.
�
3 5 10 + 2 15( ) ! ! 22.
�
3 5 + 2( ) 5 −1( )
�
distrubite the 3 5to both terms
3 5 • 10 + 3 5 • 2 15
simplify
3 5 • 2 • 5 + 3• 2 5 • 5 • 3
15 2 + 30 3
!
�
F O I L
3 5 • 5 + 3 5 • −1( ) + 2 • 5 + 2(−1)
simplify
3 25 − 3 5 + 2 5 − 2
15− 3 5 + 2 5 − 2
13− 5
24.
�
4 − 5( ) 4 + 5( ) ! !
�
F O I L
4 • 4 + 4 • 5 − 5 • 4 − 5 • 5
simplify
16 + 4 5 − 4 5 − 25
16 + 4 5 − 4 5 − 5
11
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
26.
�
46! ! `! 28.
�
145 − 7
!
�
multiply by 66
46
•66
simplify
4 636
4 66
2 63
! ! !
�
multiply by5 + 7( )5 + 7( )
145 − 7
•5 + 7( )5 + 7( )
simplify
14 5 + 7( )25 − 5 7 + 5 7 − 49
14 5 + 7( )25 − 7
14 5 + 7( )18
7 5 + 7( )9
31.
�
a = 3 5
�
c = 7 Find b.
�
a2 + b2 = c2
3 5( )2 + b2 = 72
3 5( ) 3 5( ) + b2 = 72
9 25( ) + b2 = 49
45 + b2 = 49
b2 = 4b = 2
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
Solve for x.
32.
�
5x − 6 −1= 2! 33.
�
2x + 40 + 6 = −2 ! !
Get the radical alone on one side of the equation
5x − 6 = 3
Is the constant negitive?No, then proceed.
Square both sides of the equation
5x − 6 = 9
Solve for x
5x = 15x = 3
!
Get the radical alone on one side of the equation
2x + 40 = 3− 8Is the constant negitive?Yes, then STOP.
No Real Number works
NRN
34. 3x − 6 = −2x +14 ! 35. 5x −19 = 2x + 4 !
Both radicals are alone onthier side of the equation
Square both sides of the equation
3x − 6 = −2x +14
Solve for x
5x − 6 = 145x = 20x = 4
!
Both radicals are alone onthier side of the equation
Square both sides of the equation
5x +19 = 2x + 4
Solve for x
3x +19 = 43x = −15x = −5
When you plug in x = – 5it does not workNRN or ∅
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
36. 2x2 + 3x −10 = x ! 37. 2x2 − x −1 = 2x − 2
Square both sides of the equation
2x2 + 3x −10 = x2
Solve for x
x2 + 3x −10 = 0(x + 5)(x − 2) = 0
x = −5 or x = 2
When you plug in x = 2it does workWhen you plug in x = – 5it does NOT work
Answer: x = 2
!
Square both sides of the equation
2x2 − x −1= 2x − 2
Solve for x
2x2 − 3x +1= 0(2x −1)(x −1) = 0
x = 12
or x = 1
When you plug in x = 1it does work
When you plug in x = 12
it does NOT work
Answer: x = 1
!
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
�
x =−(B)± B( )2 − 4(A)(C)
2A
Quadratic Equations. Solve for x. You must use the quadratic equation for each problem and show the work. Use NRN or
�
∅ if needed.
38.
�
3x2 + 2x − 3 = 0 ! 39.
�
4x2 −8x +1 = 0 !
�
A B C
3x2 + 2x − 3 = 0
B2 − 4 A( ) C( ) = 2( )2 − 4 3( ) −3( )= 4 + 36 = 40
x =−(2) ± 40
6
x =−2 ± 4 •10
6
x =−2 ± 2 10
6
x =2 −1± 10( )
6
x =−1± 10
3
!
�
A B C
4x2 − 8x + 1 = 0
B2 − 4 A( ) C( ) = −8( )2 − 4 4( ) 1( )= 64 −16 = 48
x =−(−8) ± 48
8
x =8 ± 3•16
8
x =8 ± 4 3
8
x =4 2 ± 3( )
8
x =2 ± 3
2
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
40.
�
2x2 + 3x + 4 = 0 !
�
A B C
2x2 + 3x + 4 = 0
B2 − 4 A( ) C( ) = 3( )2 − 4 2( ) 4( )= 9 − 32 = −23
x =−(2) ± − 23
4
Thereis no real number
for − 23so the answer is no real number
NRN
Math 100! Half Chapter 8 and Chapter 9 Test Review WKD! © 2016Eitel
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