SOLUTIONS TO ALGEBRA EXERCISES

TABLE OF CONTENTS

click a link to go to the solutions on each topic

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A/ Exponents

a)	b)	c)	d)	e)	f)
g) a + b	h) a² + b²	i) a2(1 + b3)	j) 1

2) Find a numerical value for the questions in #1 if a = -2 and b = -3.

a)	b)	c)	d)	e)
f)	g) – 5	h) 13	i) 40	j) 1

3) Simplify the following and find numerical values when possible.

a) a6 .

b) x4

c) 64

d) 27

e) – 35= – 243

f)

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B/ Factoring

4) Factor completely.

a) (h + k)(x + y)	b) (a + b)(a + b + 2)	c) (x – y)(a – b)
d) (x + z)(x – y)	e) 10x(x – 3)(x + 3)	f) 3(1 – 4x)(1 + 4x)
g) 2(a – 4b)(a + 4b)	h) (2m + p + 8q)(2m – p – 8q)	i) (5 – a + b)(5 – a – b)
j) (x + 6)(x + 1)	k) (a – 12)(a + 2)	l) 3(a + 8)(a – 1)
m) (5 – 3m)(3 + 2m)	n) (5m – 2)(m + 1)	o) (8x + y)(x – 4y)
p) (9a – 4)(2a + 3)	q) 2(x – 2y)(x2 + 2xy + 4y²)	r) (a + 5b)(a2 – 5ab + 25b²)

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C/ Rational Numbers (fractions)

5) Perform the indicated operations. Reduce to lowest terms.

a)

b)

c)

d)

e)

f)

g)

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6) Perform the indicated operations. Reduce to lowest terms.

a)

b)

c)

d)

e)

f)

g)

h)

i)

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D/ Equations

7) Solve the following equations. Check your answers.

a) 3(4x – 1) = 9 – 4x 12x –3 = 9 – 4x 16x = 12 x = ¾	b) 5x – 3(x + 1) = 2(3x + 5) 5x – 3x – 3 = 6x + 10 – 4x = 13 x = – 13/4	c) 15w – 4 = 8w + 31 7w = 35 w = 5
d) 8y2 + 2y –1 = 0 (4y – 1)(2y + 1) = 0 y = ¼ or y = – 1/2	e) 4x2 + 5cx – 9c2 = 0 (4x + 9c)(x – c) = 0 x = – 9c/4 or x = c	f) 2x2 + 2x = 3 2x2 + 2x – 3 = 0
g) 4x + 5 = 9x2 + 24x + 16 0 = 9x2 + 20x + 11 0 = (9x + 11)(x + 1) x = – 11/9 or x = – 1		h) 16x = 144 – 24x + x² x2 – 40x + 144 = 0 (x – 4)(x – 36) = 0; x = 4 or x = 36

i) 16 = y + 7 so y = 9	j) | 2x – 3 | = 5 so 2x – 3 = 5 or 2x – 3 = – 5 2x = 8 or 2x = – 2 x = 4 or x = – 1
	k) | 5x + 7 | = | 2x – 3 | 5x + 7 = 2x – 3 or 5x + 7 = – 2x + 3 3x = – 10 or 7x = – 4 x = – 10/3 or x = – 4/7
l) 3(x + 4) – 2(5) = 2x + 3 3x + 12 – 10 = 2x + 3 x = 1	m) x + 2 + 7x2 – 14x – 26 = 0 7x2 – 13x – 24 = 0 (7x + 8) (x – 3) = 0 so, x = – 8/7 or x = 3
n) 8a – 32 – a2 + 3a = a2 – 9a + 18 2a2 – 20a + 50 = 0 becomes a2 – 10a + 25 = 0 (a – 5)2 = 0 so a = 5

8) Solve these systems of equations. Check your answers.

a) x + y = 6 becomes x = 6 – y x – y = 2 becomes (6 – y) – y = 2 6 – 2y = 2 so y = 2 so x = 4	b) 2x – y = 2 becomes y = 2x – 2 2x + y = 10 becomes 2x + (2x – 2) = 10 4x = 12 so x = 3 and y = 4
c) x + 2y = 11 becomes x = 11 – 2y 3x – y = 5 becomes 3(11 – 2y) – y = 5 – 7y = – 28 so y = 4 so x = 3	d) 2x + y = 9 becomes y = 9 – 2x 3x = y + 1 becomes 3x = 9 – 2x + 1 5x = 10 so x = 2 and y = 5
e) x2 + y2 = 26 3x – 2y = 13 becomes 4y2 + 52y + 169 + 9y2 = 234 13y2 + 52y – 65 = 0 y2 + 4y – 5 = 0 (y + 5)(y – 1) = 0 so, y = -5 or y = 1 x = 1 or x = 5	f) x2 + y2 = 8 – 3x + y = 8 becomes y = 3x + 8 x2 + (3x + 8)2 = 8 x2 + 9x2 + 48x + 64 = 8 10x2 + 48x + 56 = 0 5x2 + 24x + 28 = 0 (5x + 14)(x + 2) = 0 so x = – 2 or y = 2 or
g) x2 + y2 = 25 x – y = – 1 becomes y = x – 1 x² + (x – 1)² = 25 x² + x² – 2x + 1 – 25 = 0 2x² – 2x – 24 = 0 x² – x – 12 = 0 (x – 4) (x + 3) = 0 x = 4 or x = – 3; y = 3 or y = – 4	h) y = x2 – 3x – 4 y + 6 = 0 becomes y = – 6 – 6 = x2 – 3x – 4 becomes 0 = x2 – 3x + 2 0 = (x – 1) (x – 2) x = 1 or x = 2 and y = – 6

E/ Inequalities

9) Solve these inequalities.

a) 3x – 2 < x + 6 2x < 8 so x < 4	b) 4x + 3 > 5x – 17 – x > – 20 so x < 20
c) | 3x + 1 | < 7 – 7 < 3x + 1 < 7 so – 8/3 < x < 2	d) | 5x – 4 | > 21 5x – 4 > 21 or 5x – 4 < – 21 5x > 25 or 5x < – 17 x > 5 or x < – 17/5
e) x2 – 4x < 5 x2 – 4x – 5 < 0 (x – 5)(x + 1) < 0 so – 1 < x < 5	f) 2x2 > 5x + 3 2x2 – 5x – 3 > 0 (2x + 1)(x – 3) > 0 so x < – 1/2, x > 3
g) 2x – 6 + 6x + 3 < 13 8x < 16 so x < 2	h) 90 – 40x + 5 > 6x + 3 – 46x > – 92 so x < 2
i) – 4 < x < – 2 or x > 2	j) values which make brackets = 0 are: x = – 3, 1/3, – 2, and 7 – 3 < x < – 2 or 1/3 < x < 7

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