Download PPT Review 4.6 - 4.7-0

Solve each equation or inequality
1. b - 10 = -3
b
é 10
ù
êëb- = -3úû(b)
b
b -10 = -3b
2
b + 3b-10 = 0
2
(b+ 5)(b- 2) = 0
b = 2, -5
(12)
3y
+1
2
+
4
y
5
é 3y+1
ù
4y
5
2. ê
+
==-- ú
ë 4
33
66û
3(3y+1) + 4(2 + 4y) = 2(-5)
9y+ 3+ 8 +16y = -10
25y = -21
-21
y=
25
Solve each equation or inequality
3. 2 + 6 £ -5
w w -1
a
6
4.
+
=2
a - 2 a+ 2
é2
ù
6
êë w + w -1 £ -5úû(w)(w -1)
é a
ù
6
êë a - 2 + a+ 2 = 2 úû(a - 2)(a+ 2)
2(w -1) + 6(w) £ -5(w)(w -1)
a(a + 2) + 6(a - 2) = 2(a2 - 4)
2w - 2 + 6w + 5w - 5w £ 0
a2 + 2a + 6a -12 - 2a2 + 8 = 0
5w + 3w - 2 £ 0
-a2 + 8a - 4 = 0
2
2
USE QUADRATIC FORMULA
w = 2/5, -1
What values go
on your number
lines??
2/5, -1, 0, and 1
-1 £ w < 0
2 £ w <1
5
-8 ± 64 - 4(-1)(-4)
-2
-8 ± 48 -8 ± 4 3
=
-2
-2
4±2 3
Example: Solve.
3 1 12
 
x 2 x
2 x * 3 2 x 2 x *12


x
2
x
6  x  24
 x  18
x  18
LCM: 2x
Multiply each fraction
through by the LCM
3
1 12
 
 18 2  18
3  9  12
Check your
solution!
Solve.
5x
5
 4
x 1
x 1
LCM: ?
LCM: (x+1)
5 x( x  1)
5( x  1)
 4( x  1) 
( x  1)
( x  1)
5x  4 x  4  5
5x  4 x  1
x  1
5(1)
5
 4
1 1
1 1
5
5
 4
0
0
?
Check your
solution!
No Solution!
Solve.
3x  2
6
 2
1
x2 x 4
Factor 1st!
3x  2
6

1
x  2 ( x  2)( x  2)
LCM: (x+2)(x-2)
(3x  2)( x  2)( x  2) 6( x  2)( x  2)

 ( x  2)( x  2)
( x  2)
( x  2)( x  2)
3x 2  6 x  2 x  4  6  x 2  2 x  2 x  4
3x 2  4 x  4  x 2  2
2x  4x  6  0
2
x  2x  3  0
( x  3)( x  1)  0
x  3 or x  1
2
x  3  0 or x 1  0
Check your
solutions!
Decompose this into partial fractions
5. 4 p +13p -12
p3 - p2 - 2 p
2
é 44pp +13p-12
+13p-12 AA BB
CC ù
== ++
++
ê
ú
p(p-2)(p+1)
2)(p+1) pp pp-22 p+1
p+1û
ë p(p-
( p)( p-2)( p+1)
22
4(0)2
4p2 +13p-12 = A(p- 2)(p+1)+ B(p)(p+1)+C(p)(p- 2)
What values are excluded?
x≠0
x≠2
x ≠ -1
4(0)2 +13(0) -12 = A(0 - 2)(0 +1)+ B(0)(0 +1)+C(0)(0 - 2)
-12 = -2A
Then substitute other
6
5
7
+
excluded values to find
A= 6
p p - 2 p +1
B and C
Solve each equation or inequality
6.
(
x-3+4 =6
x- 3 = 2
2
2
x - 3 = ( 2)
)
x- 3 = 4
x=7
2x - 3 = 5x + 4
7.
(
) (
2
2x - 3 =
5x + 4
)
2x- 3 = 5x+ 4
-7 = 3x
-7
x=
3
No real solution
2
Solve each equation or inequality
8.
k - 7 + k - 3 = 2 9.
(
k- 7 = 2- k-3
) (
2
k- 7 = 2- k-3
)
2
k- 7 = 4- 4 k-3 +k-3
-8 = -4 k - 3
2 = k-3
4 = k-3
k=7
3
2m-1 + 6 = 3
3
(
3
2m-1 = -3
)
3
2m-1 = (-3)
2m-1= -27
m = -13
3
Solve each equation or inequality
10.
(
3t + 7 > 7
)
2
But remember that you can have a
negative value under a radical
3t + 7 = ( 7)
3t + 7 = 49
t =14
No
3t + 7 ³ 0
-7
t³
3
2
So now test around those
values on a number line
No
-7
3
Yes
t > 14
14
11.) Find the upper and lower
bound of the zeros of:
f (x) = x3 - 3x2 - 2x -1
Upper = 4
Lower = -1
Approximate the real zeros of each
functions to the nearest tenth.
4
3
f
(x)
=
x
3x
+ 2x -1
12. f (x) = -x - 2x + 4 13.
3
x = 1.2
x = -0.9, 2.8
Approximate the real zeros of each
functions
14.f (x) = x3 + 4x2 - 3x - 5 15. f (x) = -3x4 - 5x3 + x - 2
x = -4.4, -0.9, 1.3
No real zeros
16.) Find the number of positive,
negative, and imaginary:
f (x) = x3 - x2 + 6x +1
Pos. 2 or 0
Neg. 1
17.) Find the possible rational
roots: f (x) = 2x3 - 4x2 + 5x - 3
±1,±3,±1.5,±0.5
Find the remainder of each division.
Then state whether the binomial is a
factor?
3
18. (x -14 x) ¸ (x - 5)
3
Remainder = 55
Not a factor
19.
x - 6x + 9
x-3
Remainder = 18
Not a factor
20.) Use synthetic division to
3
2
x
5x
-17x - 6
divide:
x+2
x - 7x - 3
2
Graph each equation or inequality:
21. y < 4 - x - x2
2
y
=
x
+ 4x + 4
22.
Solve each equation or inequality
23. 2x2 - 5x + 2 = 0
x=2
x=1
2
24. 3x - x +10 = 0
2
1 ± i 119
x=
6
Solve each equation by completing
the square:
25. x2 - 5x - 84 = 0
x = 12
x = -7
7
1
26. x - x + = 0
12
12
2
1
x=
3
1
x=
4
27.) Find the discriminant of the
function given and describe the
nature of the roots. 2x2 - 8 + 3x = 0
Discriminant = 73
2 distinct real roots
Write the polynomial equation of
least degree have the roots:
28.) 1, -1, 0.5
2x3 - x2 - 2x +1 = 0
Find the roots of each equation:
29. x + 36 = 0
2
x = ±6i
30. 4x -10x - 24x = 0
3
2
x = 0,4,-1.5