Review 4.6-4.7

Solve each equation or inequality

1.

€

b −10

b= −3 2.

€

3y +1

4+

2 + 4y

3= −

5

6

b = 2, -5

€

y =−21

25

Multiply every term by 12

€

3(3y +1) + 4(2 + 4y) = 2(−5)

9y + 3+ 8 +16y = −10

25y = −21

€

b −10

b= −3

⎡ ⎣ ⎢

⎤ ⎦ ⎥(b)

b2 −10 = −3b

b2 + 3b −10 = 0

Solve each equation or inequality

3.

€

2

w+

6

w −1≤ −5 4.

€

a

a − 2+

6

a+ 2= 2

€

−1 ≤ w < 0

25 ≤ w <1

€

4 ± 2 3

€

€

2

w+

6

w −1≤ −5

⎡ ⎣ ⎢

⎤ ⎦ ⎥(w)(w −1)

€

2(w −1) + 6(w) ≤ −5(w)(w −1)

2w − 2 + 6w + 5w2 − 5w ≤ 0

5w2 + 3w − 2 ≤ 0w = 2/5, -1

€

a

a − 2+

6

a+ 2= 2

⎡ ⎣ ⎢

⎤ ⎦ ⎥(a − 2)(a+ 2)

€

a(a+ 2) + 6(a − 2) = 2(a2 − 4)

a2 + 2a+ 6a −12 − 2a2 + 8 = 0

−a2 + 8a − 4 = 0

USE QUADRATIC FORMULA

€

−8 ± 64 − 4(−1)(−4)

−2What values go on your number lines??

2/5, -1, 0, and 1

€

−8 ± 48

−2=

−8 ± 4 3

−2

Example: Solve.

xx

12

2

13

LCM: 2x

Multiply each fraction through by

the LCM

x

xx

x

x 12*2

2

23*2

246 x

18 x

18xCheck your solution!

18

12

2

1

18

3

1293

Solve.1

54

1

5

xx

x LCM: ?LCM: (x+1)

)1(

)1(5)1(4

)1(

)1(5

x

xx

x

xx

5445 xx

145 xx

1x

Check your solution!

11

54

11

)1(5

0

54

0

5

?

No Solution!

Solve. 14

6

2

232

xx

x

Factor 1st!

1)2)(2(

6

2

23

xxx

x

LCM: (x+2)(x-2)

)2)(2()2)(2(

)2)(2(6

)2(

)2)(2)(23(

xx

xx

xx

x

xxx

42264263 22 xxxxxx

2443 22 xxx

0642 2 xx0322 xx

0)1)(3( xx

01or 03 xx

1or 3 xx

Check your solutions!

Example: Solve.

4

1

4

32

xxx

xx 42 12342 xxx

0122 xx0)3)(4( xx

03or 04 xx3or 4 xx

Check your solutions!

)4(3 x

Last Example: Solve. 1

2

22

62

x

x

xx

)1(6 x)2)(1(2 xxx

6)2(2 xx3)2( xx

0322 xx0)1)(3( xx

01or 03 xx1or 3 xx

1

2

)1(2

6

x

x

xx

Check your solutions!

Solve Check your solution.

The LCD for the three denominators is

Original equation

Multiply each sideby 24(3 – x).

1 1

11 1

6

Simplify.

Simplify.

Add.

Check Original equation

Simplify.

Simplify.

The solution is correct.

Answer: The solution is –45.

Answer:

Solve

Solve Check your solution.

The LCD is

Original equation

Multiply by the LCD, (p2 – 1).

p – 1

1

1

1

DistributiveProperty

Simplify.

Simplify.

Add(2p2 – 2p + 1)to each side.

Factor.

or Zero ProductProperty

Solve eachequation.

Divide eachside by 3.

Check Original equation

Simplify.

Simplify.

Since p = –1 results in a zero in the denominator, eliminate –1.

Answer: The solution is p = 2.

Simplify.

Original equation

Answer:

Solve

Decompose this into partial fractions

5.

€

4 p2 +13p −12

p3 − p2 − 2p

€

6

p+

5

p − 2−

7

p+1

Solve

Step 1 Values that make the denominator equal to 0 are excluded from the denominator. For this inequality the excluded value is 0.

Step 2 Solve the related equation.

Related equation

Multiply each side by 9s.

Simplify.

Add.

Divide each side by 6.

Step 3 Draw vertical lines at the excluded value and at the solution to separate the number line into regions.

Now test a sample value in each region to determine if the values in the region satisfy the inequality.

Test

is a solution.

is not a solution.

Test

is a solution.

Test

Answer: The solution

Solve

Answer:

Solve each equation or inequality

6.

€

x − 3 + 4 = 6 7.

€

2x − 3 = 5x + 4

x = 7 No real solution

Solve each equation or inequality

8.

€

k − 7 + k − 3 = 2 9.

€

2m −13 + 6 = 3

k = 7 m = -13

Solve each equation or inequality

10.

€

3t + 7 > 7

t > 14

11.) Find the upper and lower bound of the zeros of:

€

f (x) = x 3 − 3x 2 − 2x −1

Upper = 4Lower = -1

Approximate the real zeros of each functions

12.

€

f (x) = −x 3 − 2x + 4 13.

€

f (x) = x 4 − 3x 3 + 2x −1

Between 1 and 2 Between -1 and 0Between 2 and 3

Approximate the real zeros of each functions

14.

€

f (x) = x 3 + 4x 2 − 3x − 5 15.

€

f (x) = −3x 4 − 5x 3 + x − 2

Between -5 and -4, -1 and 0, 1 and 2 No real zeros

16.) Find the number of positive, negative, and imaginary:

€

f (x) = x 3 − x 2 + 6x +1

Pos. 2 or 0Neg. 1

17.) Find the possible rational roots:

€

f (x) = 2x 3 − 4x 2 + 5x − 3

€

±1,±3,±1.5,±0.5

Find the remainder of each division.Then state whether the binomial is a factor?18.

€

(x 3 −14x) ÷ (x − 5) 19.

€

x 3 − 6x + 9

x − 3

Remainder = 55Not a factor

Remainder = 18Not a factor

20.) Use synthetic division to divide:

€

x 3 − 5x 2 −17x − 6

x + 2

€

x 2 − 7x − 3

Graph each equation or inequality:

21.

€

y < 4 − x − x 2 22.

€

y = x 2 + 4x + 4

Solve each equation or inequality23.

€

2x 2 − 5x + 2 = 0 24.

€

3x 2 − x +10 = 0

€

x = 2

x = 12

€

x =1± i 119

6

Solve each equation by completing the square:

25.

€

x 2 − 5x − 84 = 0 26.

€

x 2 −7

12x +

1

12= 0

€

x =12

x = −7

€

x =1

3

x =1

4

27.) Find the discriminant of the function given and describe the nature of the roots.

Discriminant = 732 distinct real roots

€

2x 2 − 8 + 3x = 0

Write the polynomial equation of least degree have the roots:

28.) 1, -1, 0.5

€

2x 3 − x 2 − 2x +1 = 0

Find the roots of each equation:

29.

€

x 2 + 36 = 0 30.

€

4x 3 −10x 2 − 24x = 0

€

x = ±6i

€

x = 0,4,−1.5