Ass_MTH100_2333

1

MTH 100 Spring 2015

Dr. Yassine

Name: Ahmed Al Qubaisi I.D. No. S0000001654

Assignment 2

Deadline to bring the assignment is 4th

December, 2015

Exercise 1 (10pt). Quadratic Equations Find all possible values of the given variable.

1.

Simplifying

(4y + -8)(3y + 7) = 0

Reorder the terms:

(-8 + 4y)(3y + 7) = 0

Reorder the terms:

(-8 + 4y)(7 + 3y) = 0

Multiply (-8 + 4y) * (7 + 3y)

2

(-8(7 + 3y) + 4y * (7 + 3y)) = 0

((7 * -8 + 3y * -8) + 4y * (7 + 3y)) = 0

((-56 + -24y) + 4y * (7 + 3y)) = 0

(-56 + -24y + (7 * 4y + 3y * 4y)) = 0

(-56 + -24y + (28y + 12y2)) = 0

Combine like terms: -24y + 28y = 4y

(-56 + 4y + 12y2) = 0

Solving

-56 + 4y + 12y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '4'.

4(-14 + y + 3y2) = 0

Factor a trinomial.

4((-7 + -3y)(2 + -1y)) = 0

Ignore the factor 4.

Subproblem 1

Set the factor '(-7 + -3y)' equal to zero and attempt to solve: Simplifying -7 + -3y = 0 Solving -7 + -3y = 0 Move all terms containing y to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + -3y = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -3y = 0 + 7 -3y = 0 + 7

3

Combine like terms: 0 + 7 = 7 -3y = 7 Divide each side by '-3'. y = -2.333333333 Simplifying y = -2.333333333

2.

Simplifying

(4w + -8)(w + 3) = 0

Reorder the terms:

(-8 + 4w)(w + 3) = 0

Reorder the terms:

(-8 + 4w)(3 + w) = 0

Multiply (-8 + 4w) * (3 + w)

(-8(3 + w) + 4w * (3 + w)) = 0

((3 * -8 + w * -8) + 4w * (3 + w)) = 0

((-24 + -8w) + 4w * (3 + w)) = 0

(-24 + -8w + (3 * 4w + w * 4w)) = 0

(-24 + -8w + (12w + 4w2)) = 0

Combine like terms: -8w + 12w = 4w

(-24 + 4w + 4w2) = 0

Solving

-24 + 4w + 4w2 = 0

4

Solving for variable 'w'.


4(-6 + w + w2) = 0

Factor a trinomial.

4((-3 + -1w)(2 + -1w)) = 0


Subproblem 1

Set the factor '(-3 + -1w)' equal to zero and attempt to solve: Simplifying -3 + -1w = 0 Solving -3 + -1w = 0 Move all terms containing w to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + -1w = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1w = 0 + 3 -1w = 0 + 3 Combine like terms: 0 + 3 = 3 -1w = 3 Divide each side by '-1'. w = -3 Simplifying w = -3

5

3.

Simplifying

(p + -5)(p + 2) = 0

Reorder the terms:

(-5 + p)(p + 2) = 0

Reorder the terms:

(-5 + p)(2 + p) = 0

Multiply (-5 + p) * (2 + p)

(-5(2 + p) + p(2 + p)) = 0

((2 * -5 + p * -5) + p(2 + p)) = 0

((-10 + -5p) + p(2 + p)) = 0

(-10 + -5p + (2 * p + p * p)) = 0

(-10 + -5p + (2p + p2)) = 0

Combine like terms: -5p + 2p = -3p

(-10 + -3p + p2) = 0

Solving

-10 + -3p + p2 = 0

6

Solving for variable 'p'.

Factor a trinomial.

(-2 + -1p)(5 + -1p) = 0

Subproblem 1

Set the factor '(-2 + -1p)' equal to zero and attempt to solve: Simplifying -2 + -1p = 0 Solving -2 + -1p = 0 Move all terms containing p to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + -1p = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1p = 0 + 2 -1p = 0 + 2 Combine like terms: 0 + 2 = 2 -1p = 2 Divide each side by '-1'. p = -2 Simplifying p = -2

7

4.

Simplifying

(k + 4)(3k + -12) = 0

Reorder the terms:

(4 + k)(3k + -12) = 0

Reorder the terms:

(4 + k)(-12 + 3k) = 0

Multiply (4 + k) * (-12 + 3k)

(4(-12 + 3k) + k(-12 + 3k)) = 0

((-12 * 4 + 3k * 4) + k(-12 + 3k)) = 0

((-48 + 12k) + k(-12 + 3k)) = 0

(-48 + 12k + (-12 * k + 3k * k)) = 0

(-48 + 12k + (-12k + 3k2)) = 0

Combine like terms: 12k + -12k = 0

(-48 + 0 + 3k2) = 0

(-48 + 3k2) = 0

Solving

-48 + 3k2 = 0

Solving for variable 'k'.

8

Move all terms containing k to the left, all other terms to the right.

Add '48' to each side of the equation.

-48 + 48 + 3k2 = 0 + 48

Combine like terms: -48 + 48 = 0

0 + 3k2 = 0 + 48

3k2 = 0 + 48

Combine like terms: 0 + 48 = 48

3k2 = 48

Divide each side by '3'.

k2 = 16

Simplifying

k2 = 16

Take the square root of each side:

k = {-4, 4}

9

5.

Simplifying

(m + 11)(m + -9) = 0

Reorder the terms:

(11 + m)(m + -9) = 0

Reorder the terms:

(11 + m)(-9 + m) = 0

Multiply (11 + m) * (-9 + m)

(11(-9 + m) + m(-9 + m)) = 0

((-9 * 11 + m * 11) + m(-9 + m)) = 0

((-99 + 11m) + m(-9 + m)) = 0

(-99 + 11m + (-9 * m + m * m)) = 0

(-99 + 11m + (-9m + m2)) = 0

Combine like terms: 11m + -9m = 2m

(-99 + 2m + m2) = 0

Solving

-99 + 2m + m2 = 0

Solving for variable 'm'.

Factor a trinomial.

(-11 + -1m)(9 + -1m) = 0

10

Subproblem 1

Set the factor '(-11 + -1m)' equal to zero and attempt to solve: Simplifying -11 + -1m = 0 Solving -11 + -1m = 0 Move all terms containing m to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + -1m = 0 + 11 Combine like terms: -11 + 11 = 0 0 + -1m = 0 + 11 -1m = 0 + 11 Combine like terms: 0 + 11 = 11 -1m = 11 Divide each side by '-1'. m = -11 Simplifying m = -11

Subproblem 2

Set the factor '(9 + -1m)' equal to zero and attempt to solve: Simplifying 9 + -1m = 0 Solving 9 + -1m = 0 Move all terms containing m to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + -1m = 0 + -9 Combine like terms: 9 + -9 = 0 0 + -1m = 0 + -9 -1m = 0 + -9 Combine like terms: 0 + -9 = -9 -1m = -9

11

Divide each side by '-1'. m = 9 Simplifying m = 9

Solution

m = {-11, 9}

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6.

Simplifying

(4k + -2)(3k + 6) = 0

Reorder the terms:

(-2 + 4k)(3k + 6) = 0

Reorder the terms:

(-2 + 4k)(6 + 3k) = 0

Multiply (-2 + 4k) * (6 + 3k)

(-2(6 + 3k) + 4k * (6 + 3k)) = 0

((6 * -2 + 3k * -2) + 4k * (6 + 3k)) = 0

((-12 + -6k) + 4k * (6 + 3k)) = 0

(-12 + -6k + (6 * 4k + 3k * 4k)) = 0

(-12 + -6k + (24k + 12k2)) = 0

Combine like terms: -6k + 24k = 18k

(-12 + 18k + 12k2) = 0

Solving

-12 + 18k + 12k2 = 0

Solving for variable 'k'.


6(-2 + 3k + 2k2) = 0

Factor a trinomial.

13

6((-2 + -1k)(1 + -2k)) = 0


Subproblem 1

Set the factor '(-2 + -1k)' equal to zero and attempt to solve: Simplifying -2 + -1k = 0 Solving -2 + -1k = 0 Move all terms containing k to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + -1k = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1k = 0 + 2 -1k = 0 + 2 Combine like terms: 0 + 2 = 2 -1k = 2 Divide each side by '-1'. k = -2 Simplifying k = -2

Subproblem 2

Set the factor '(1 + -2k)' equal to zero and attempt to solve: Simplifying 1 + -2k = 0 Solving 1 + -2k = 0 Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + -2k = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2k = 0 + -1 -2k = 0 + -1

14

Combine like terms: 0 + -1 = -1 -2k = -1 Divide each side by '-2'. k = 0.5 Simplifying k = 0.5

Solution

k = {-2, 0.5}

7.

Simplifying

(3h + -7)(5h + 4) = 0

Reorder the terms:

(-7 + 3h)(5h + 4) = 0

Reorder the terms:

(-7 + 3h)(4 + 5h) = 0

Multiply (-7 + 3h) * (4 + 5h)

(-7(4 + 5h) + 3h * (4 + 5h)) = 0

((4 * -7 + 5h * -7) + 3h * (4 + 5h)) = 0

((-28 + -35h) + 3h * (4 + 5h)) = 0

(-28 + -35h + (4 * 3h + 5h * 3h)) = 0

15

(-28 + -35h + (12h + 15h2)) = 0

Combine like terms: -35h + 12h = -23h

(-28 + -23h + 15h2) = 0

Solving

-28 + -23h + 15h2 = 0

Solving for variable 'h'.

Factor a trinomial.

(-4 + -5h)(7 + -3h) = 0

Subproblem 1

Set the factor '(-4 + -5h)' equal to zero and attempt to solve: Simplifying -4 + -5h = 0 Solving -4 + -5h = 0 Move all terms containing h to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + -5h = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -5h = 0 + 4 -5h = 0 + 4 Combine like terms: 0 + 4 = 4 -5h = 4 Divide each side by '-5'. h = -0.8 Simplifying h = -0.8

Subproblem 2

Set the factor '(7 + -3h)' equal to zero and attempt to solve:

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Simplifying 7 + -3h = 0 Solving 7 + -3h = 0 Move all terms containing h to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + -3h = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -3h = 0 + -7 -3h = 0 + -7 Combine like terms: 0 + -7 = -7 -3h = -7 Divide each side by '-3'. h = 2.333333333 Simplifying h = 2.333333333

Solution

h = {-0.8, 2.333333333}

8.

Simplifying

17

(6m + -7)(2m + 12) = 0

Reorder the terms:

(-7 + 6m)(2m + 12) = 0

Reorder the terms:

(-7 + 6m)(12 + 2m) = 0

Multiply (-7 + 6m) * (12 + 2m)

(-7(12 + 2m) + 6m * (12 + 2m)) = 0

((12 * -7 + 2m * -7) + 6m * (12 + 2m)) = 0

((-84 + -14m) + 6m * (12 + 2m)) = 0

(-84 + -14m + (12 * 6m + 2m * 6m)) = 0

(-84 + -14m + (72m + 12m2)) = 0

Combine like terms: -14m + 72m = 58m

(-84 + 58m + 12m2) = 0

Solving

-84 + 58m + 12m2 = 0

Solving for variable 'm'.


2(-42 + 29m + 6m2) = 0

Factor a trinomial.

2((-6 + -1m)(7 + -6m)) = 0


Subproblem 1

Set the factor '(-6 + -1m)' equal to zero and attempt to solve: Simplifying

18

-6 + -1m = 0 Solving -6 + -1m = 0 Move all terms containing m to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + -1m = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -1m = 0 + 6 -1m = 0 + 6 Combine like terms: 0 + 6 = 6 -1m = 6 Divide each side by '-1'. m = -6 Simplifying m = -6

Subproblem 2

Set the factor '(7 + -6m)' equal to zero and attempt to solve: Simplifying 7 + -6m = 0 Solving 7 + -6m = 0 Move all terms containing m to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + -6m = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -6m = 0 + -7 -6m = 0 + -7 Combine like terms: 0 + -7 = -7 -6m = -7 Divide each side by '-6'. m = 1.166666667 Simplifying m = 1.166666667

Solution

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m = {-6, 1.166666667}

9.

Simplifying

(s + 1)(4s + -8) = 0

Reorder the terms:

(1 + s)(4s + -8) = 0

Reorder the terms:

(1 + s)(-8 + 4s) = 0

Multiply (1 + s) * (-8 + 4s)

(1(-8 + 4s) + s(-8 + 4s)) = 0

((-8 * 1 + 4s * 1) + s(-8 + 4s)) = 0

((-8 + 4s) + s(-8 + 4s)) = 0

(-8 + 4s + (-8 * s + 4s * s)) = 0

(-8 + 4s + (-8s + 4s2)) = 0

Combine like terms: 4s + -8s = -4s

(-8 + -4s + 4s2) = 0

Solving

-8 + -4s + 4s2 = 0

20

Solving for variable 's'.


4(-2 + -1s + s2) = 0

Factor a trinomial.

4((-1 + -1s)(2 + -1s)) = 0


Subproblem 1

Set the factor '(-1 + -1s)' equal to zero and attempt to solve: Simplifying -1 + -1s = 0 Solving -1 + -1s = 0 Move all terms containing s to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + -1s = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1s = 0 + 1 -1s = 0 + 1 Combine like terms: 0 + 1 = 1 -1s = 1 Divide each side by '-1'. s = -1 Simplifying s = -1

Subproblem 2

Set the factor '(2 + -1s)' equal to zero and attempt to solve: Simplifying 2 + -1s = 0 Solving 2 + -1s = 0

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Move all terms containing s to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + -1s = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -1s = 0 + -2 -1s = 0 + -2 Combine like terms: 0 + -2 = -2 -1s = -2 Divide each side by '-1'. s = 2 Simplifying s = 2

Solution

s = {-1, 2}

22

10.

Simplifying

(2z + 7)(z + 1) = 0

Reorder the terms:

(7 + 2z)(z + 1) = 0

Reorder the terms:

(7 + 2z)(1 + z) = 0

Multiply (7 + 2z) * (1 + z)

(7(1 + z) + 2z * (1 + z)) = 0

((1 * 7 + z * 7) + 2z * (1 + z)) = 0

((7 + 7z) + 2z * (1 + z)) = 0

(7 + 7z + (1 * 2z + z * 2z)) = 0

(7 + 7z + (2z + 2z2)) = 0

Combine like terms: 7z + 2z = 9z

(7 + 9z + 2z2) = 0

Solving

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7 + 9z + 2z2 = 0

Solving for variable 'z'.

Factor a trinomial.

(7 + 2z)(1 + z) = 0

Subproblem 1

Set the factor '(7 + 2z)' equal to zero and attempt to solve: Simplifying 7 + 2z = 0 Solving 7 + 2z = 0 Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + 2z = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 2z = 0 + -7 2z = 0 + -7 Combine like terms: 0 + -7 = -7 2z = -7 Divide each side by '2'. z = -3.5 Simplifying z = -3.5

Subproblem 2

Set the factor '(1 + z)' equal to zero and attempt to solve: Simplifying 1 + z = 0 Solving 1 + z = 0 Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation.

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1 + -1 + z = 0 + -1 Combine like terms: 1 + -1 = 0 0 + z = 0 + -1 z = 0 + -1 Combine like terms: 0 + -1 = -1 z = -1 Simplifying z = -1

Solution

z = {-3.5, -1}

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Exercise 2 (6pt). Rational Equations by Cross Multiplying Solve for in each of following equations.

1.

x in (-oo:+oo) x/9 = 7/3 // - 7/3 x/9-(7/3) = 0 x/9-7/3 = 0 1/9*x-7/3 = 0 // + 7/3 1/9*x = 7/3 // : 1/9 x = 7/3/1/9 x = 21 x = 21

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Simplifying

(3x + 1) = (5x + -5)

Reorder the terms:

(1 + 3x) = (5x + -5)

Remove parenthesis around (1 + 3x)

1 + 3x = (5x + -5)

Reorder the terms:

1 + 3x = (-5 + 5x)

Remove parenthesis around (-5 + 5x)

1 + 3x = -5 + 5x

Solving

1 + 3x = -5 + 5x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-5x' to each side of the equation.

1 + 3x + -5x = -5 + 5x + -5x

Combine like terms: 3x + -5x = -2x

1 + -2x = -5 + 5x + -5x

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Combine like terms: 5x + -5x = 0

1 + -2x = -5 + 0

1 + -2x = -5

Add '-1' to each side of the equation.

1 + -1 + -2x = -5 + -1

Combine like terms: 1 + -1 = 0

0 + -2x = -5 + -1

-2x = -5 + -1

Combine like terms: -5 + -1 = -6

-2x = -6

Divide each side by '-2'.

x = 3

Simplifying

x = 3

28

12 = 9x -3 =9x =15

X = 15/9

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Simplifying

(5x + -30) = (-2x + -2)

Reorder the terms:

(-30 + 5x) = (-2x + -2)

Remove parenthesis around (-30 + 5x)

-30 + 5x = (-2x + -2)

Reorder the terms:

-30 + 5x = (-2 + -2x)

Remove parenthesis around (-2 + -2x)

-30 + 5x = -2 + -2x

Solving

-30 + 5x = -2 + -2x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2x' to each side of the equation.

-30 + 5x + 2x = -2 + -2x + 2x

Combine like terms: 5x + 2x = 7x

-30 + 7x = -2 + -2x + 2x

Combine like terms: -2x + 2x = 0

30

-30 + 7x = -2 + 0

-30 + 7x = -2

Add '30' to each side of the equation.

-30 + 30 + 7x = -2 + 30


0 + 7x = -2 + 30

7x = -2 + 30


7x = 28

Divide each side by '7'.

x = 4

Simplifying

x = 4

Exercise 3 (4pt). Inverse functions

1. If , what is the value of the reciprocal of

the reciprocal = 6/5

2. If the reciprocal of is , what is the value of ? 3y -7 = 1/2 3y = 6.5 Y = 6.5/3

3. The difference between the reciprocal of and the reciprocal of is equal to the reciprocal of .

What is the value of ?

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First we write their reciprocals x: 1/x 3x: 1/3x 6: 1/6 It says the difference of 1/x and 1/3x is 1/6 so we write the eq: 1/x - 1/3x = 1/6 The lcd of x & 3x is 3x. Rewrite in 1 fraction (3-1/)3x = 1/6 2/3x = 1/6 multiply 3x on both sides to cancel out 3x on left side 3x(2/3x) = 3x(1/6) 2 = 3x/6 lowest term: 2 = x/2 multiply both sides by 2: 2(2) = 2 (x/2) x=4 Checking: 1/4 - 1/12 = 1/6

4. If the reciprocal of is , what is the value of ? 1-2x = 1/3 -2x = - 2/3 X = 2/6

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