Chapter 0 Review

Calculus

Find the inequality that represents the graphed numbers.

-6 < x < 5

Now write the interval notation: (-6, 5)

What is the interval notation of 3 < x ≤ 5

(3, 5]

Identify the notation that denotes the

statement that “x is greater than 3 and no

greater than 13”(3, 13]

Solve and graph:

]

Solve and graph:

)

Test around pts. 6 and -4

(

(-∞, -4) U (6, ∞)

Solve and graph:

](

(-3, -1]

Adrienne is planning a 4-hour hike, and is looking for a park within a reasonable distance from her house. She can drive at an average of 35 miles per hour, but she does not want to spend more than 6 hours away from home. Which describes the possible one-way distance Adrienne can travel from her home to the park?

Sean works weekends and earns $5.30 per hour after deductions. He wants at least $335 for a stereo system. What is the fewest hours he must work to reach this goal?

64 hours

The force F applied to an elevator cable by the total weight s of the elevator can be determined by , where F is in newtons. The sign in the elevator limits the total weight of passengers and baggage to 2,500 kilograms. The empty elevator weighs 1,100 kilograms. What inequality expresses the possible range of the force on the elevator cable in newtons (N)??

A company’s monthly cost C, in dollars, for storing x surplus units of a product is:

Find the widest range of x values for which the monthly cost will not exceed $4,400

Katalin is on a mountain 11,033 feet above sea level. Nick is in a submarine 3414 feet below sea level. Which of the following can be used to find the difference between Katalin’s elevation and Nick’s elevation?

Find the distance between a and b, a = -9.7, b = 2

11.7

Find the directed distance from a to b, a = -9.2, b = -3.5

5.7

Write an absolute value inequality from the given information.

(3, 8)

Write an absolute value inequality from the given information.

(-∞, -3) U (4, ∞)

Mr. Williams sells jumbo-size bags of peanut butter chocolate chunk cookies. The number of cookies in each bag must not differ from 100 by more than 7 cookies. Find an inequality which describes b, the acceptable number of cookies in each bag.

For a cupboard door to meet specifications at a carpentry shop, the width must be within ⅛ inch of the expected width of the door. Find an inequality that expresses the range of acceptable widths for doors that are 2 feet wide, and find the minimum acceptable width of the doors.

Find the midpoint of the given interval:

[2, 7]

Evaluate when x = -2

Evaluate when x = 3

Evaluate when x = 81

Evaluate when x = 9

Simplify:

Simplify:

Simplify:

Simplify:

Find the domain of the given expression.

Find the domain of the given expression.

Find the domain of the given expression.

Test around pts. and

Find the domain of the given expression.

Test around pts. and

Find the complete factorization of the polynomial.

Find the factors of 72 that add up to 22

18 and 4

Break up the middle term

Find the complete factorization of the polynomial.

Find the complete factorization of the polynomial.

Factoring doesn’t work, so let’s try synthetic division.Try all the possible rational roots.

Find the complete factorization of the polynomial.

Sum of Cubes

Find the complete factorization of the polynomial.

Difference of Squares

Find the interval on which the given expression is defined.

Can’t have negative value under the square root sign

Test points around -4 and -3

(-∞, -4] U [-3, ∞)

Find the interval on which the given expression is defined.

Can’t have negative value under the square root sign

Test points around -6 and -5

(-∞, -6] U [-5, ∞)

Find the domain of the given expression

Can’t have negative value under the square root sign

Test points around 2 and 5

[2, 5]

Find the interval on which the given expression is defined.

Can’t have negative value under the square root sign

Test points around

Use synthetic division to complete the indicated factorization.

Use synthetic division to complete the indicated factorization.

Use synthetic division to complete the indicated factorization.

Use synthetic division to complete the indicated factorization.

Use the Rational Zero Theorem to determine all possible rational zeros of the polynomial. Do not find the actual zeros.

PQ

FACTORS OF “P”: ±1, ±3

FACTORS OF “Q”: ±1, ±3

“P over Q”

Use the Rational Zero Theorem to determine all possible rational zeros of the polynomial. Do not find the actual zeros.

PQ

FACTORS OF “P”: ±1, ±2, ±3, ±6

FACTORS OF “Q”: ±1, ±3

“P over Q”

Use the rational zero theorem as an aid in finding all the real zeros of the polynomial.

FACTORS OF “P”: ±1, ±2, ±4, ±5, ±10, ±20

FACTORS OF “Q”: ±1

So 1 is a root

So 1, -5, and 4 are roots

Use the rational zero theorem as an aid in finding all the real zeros of the polynomial.

FACTORS OF “P”: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30

FACTORS OF “Q”: ±1

So -2 is a root

So 1, 5, and -3 are roots

Simplify a Rational Expression

1

)2(

)2(1

)2(

)2(1x-2

2-x :Simplify

x

x

x

x Factor the numerator and denominator.

Divide both the numerator and denominator by the common factor, 2-x.

The numerator and denominator are opposites, or additive inverses. They differ

only in their signs.

Factors that are Opposites

EXAMPLEEXAMPLE

Simplifying Rational Expressions

EXAMPLEEXAMPLE

Simplify: .32

12

xx

x

SOLUTIONSOLUTION

Factor the numerator and denominator.

13

11

32

12

xx

x

xx

x

Divide out the common factor, x + 1.

13

11

xx

x

3

1

x

Simplify.

Simplifying Rational Expressions

EXAMPLEEXAMPLE

Simplify: .9

1272

2

x

xx

SOLUTIONSOLUTION

Factor the numerator and denominator.

Rewrite 3 – x as (-1)(-3 + x).

xx

xx

x

xx

33

43

9

1272

2

xx

xx

313

43

313

43

xx

xxRewrite -3 + x as x – 3.

Simplifying Rational Expressions

Divide out the common factor, x – 3.

Simplify.

313

43

xx

xx

CONTINUECONTINUEDD

13

4

x

x

2

6

)4(2

)6)(4(

)4(2

)6)(4(8-2x

24-2x x :Simplify

2

x

x

xx

x

xx Factor the numerator and denominator.

Divide both the numerator and denominator by the common factor, x-4.

Simplifying Rational Expressions

EXAMPLEEXAMPLE

Simplify

Assume the denominator cannot equal zero.

Factor out a negative one to move things around

ba

ba

baa

aba

2

2

2

23

4

33

44

Simplify

Assume the denominator cannot equal zero.

ba

ba

ba

ba

baa

aba

2

2

2

2

2

2

3

4

3

4

33

44

Simplify

Assume the denominator cannot equal zero.

a

a

a

ba

ba

ba

ba

baa

aba

3

4

3

4

3

4

33

44

2

2

2

2

2

2

Multiply the fractions

Reduce before multiply.

34

1

1

32

2

kk

k

k

k

Multiply the fractions

Reduce before multiply.

13

1

1

3

34

1

1

3

2

2

2

kk

k

k

k

kk

k

k

k

Multiply the fractions

Reduce before multiply.

1

1

1

1

1

13

111

1

3

13

1

1

3

34

1

1

3

2

2

2

kk

kk

k

k

kk

k

k

k

kk

k

k

k

Simplify the Complex fraction

Remember fractions are division statements.

xyx

yxx

32

493

22

2

Simplify the Complex fraction

Remember fractions are division statements.

xy

x

yx

x

xyx

yxx

3249

32

49 3

22

2

3

22

2

Simplify the Complex fraction

322

23

22

2 32

493249 x

xy

yx

x

xy

x

yx

x

Simplify the Complex fraction

322

23

22

2 32

493249 x

xy

yx

x

xy

x

yx

x

3

2 231

2323 x

yx

yxyx

x

Simplify the Complex fraction

322

23

22

2 32

493249 x

xy

yx

x

xy

x

yx

x

3

2 231

2323 x

yx

yxyx

x

yxxxyx 23

11

23

1

Simplify this expression:

Simplify this expression:

Simplify this expression:

Simplify this expression:

Simplify this expression:

Simplify this expression:

Simplify this expression:

Simplify this expression:

???? Missing a step?????

Rationalize the denominator:

Rationalize the numerator:

Rationalize the numerator:

Rationalize the numerator:

The Whole Process

Rewrite numeratorsMake new denominators using LCDMake equivalent expressionsCombine Expressions over One Denominator

3x2 2x5

2 5

2 56x5 6x5

(2x3) (3)

4x3 156x5 6x5

+

+

+

6x5

4x3 + 15

4a2b 10ab3

5 3

5 320a2b3 20a2b3

(5b2) (2a)

20a2b3 20a2b3

25b2 6a

-

-

-

20a2b3

25b2 – 6a

52c +

16c

5 + 16c 6c

(3)

166c

83c

Polynomial ProblemsFactor all polynomialsLCD is all the different “numbers” present

x2 – 4 x + 23 _ 1

(x + 2)(x – 2) (x + 2)

3 _ 1

3 _ 1

(x + 2)(x – 2) (x + 2)(x – 2)

(x – 2)

(x + 2)(x – 2)3 – 1(x – 2)

(x + 2)(x – 2)-x + 5

4 _ 53 x

4 _ 53x 3x

x (3)

3x 3x4x _ 15

4x – 15 3x

x + 2 4 x – 3

+

+(x + 2) 4

x – 3 x – 3(x – 3)

x – 3 x – 3+x2 – x – 6 4

x2 – x – 2 x – 3

(x + 1)(x – 2) x – 3

a2 – 36 6 – a+

3 2

(a + 6)(a – 6) – 1(a – 6) +

3 2

3 _ 2

(a + 6)(a – 6) (a + 6)(a – 6)

(a + 6)

(a + 6)(a – 6)3 – 2a – 12

(a + 6)(a – 6)

- 2a - 9

3x _ 5x x9x + 2 2 – 9x 81x2 – 4

+

3x 5x x

9x + 2 -1(9x – 2) (9x + 2)(9x – 2)_ +

3x 5x x+ +(9x + 2)(9x – 2) (9x + 2)(9x – 2) (9x + 2)(9x – 2)

(9x – 2) (9x + 2)

27x2 – 6x + 45x2 + 10x + x(9x + 2)(9x – 2)

(9x + 2)(9x – 2)

72x2 + 5x