SOLUTIONS TO ALGEBRA EXERCISES 
TABLE OF CONTENTS
click a link to go to the solutions on each topic
A/ Exponents 
B/ Factoring 
C/ Rational Numbers (fractions) 
D/ Equations 
E/ Inequalities 
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a)  b)  c)  d)  e)  f) 
g) a + b  h) a² + b²  i) a^{2}(1 + b^{3})  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) a^{6} .  b) x^{4}  c) 64  d) 27  e) – 3^{5}= – 243  f) 
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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)(x^{2} + 2xy + 4y²)  r) (a + 5b)(a^{2} – 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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7) Solve the following equations. Check your answers.
a) 3(4x – 1) = 9 – 4x
16x = 12 x = ¾ 
b) 5x – 3(x + 1) = 2(3x + 5)
– 4x = 13 x = – 13/4 
c) 15w – 4 = 8w + 31
w = 5 
d) 8y^{2} + 2y –1 = 0 (4y – 1)(2y + 1) = 0 y = ¼ or y = – 1/2 
e) 4x^{2} + 5cx – 9c^{2} = 0 (4x + 9c)(x – c) = 0 x = – 9c/4 or x = c 
f) 2x^{2} + 2x = 3 2x^{2} + 2x – 3 = 0

g) 4x + 5 = 9x^{2} + 24x + 16 0 = 9x^{2} + 20x + 11 0 = (9x + 11)(x + 1) x = – 11/9 or x = – 1 
h)

i)

j)  2x – 3  = 5 so 2x – 3 = 5 or 2x – 3 = – 5 2x = 8 or 2x = – 2

k)  5x + 7  =  2x – 3  5x + 7 = 2x – 3 or 5x + 7 = – 2x + 3  
l) 3(x + 4) – 2(5) = 2x + 3 
m)
x + 2 + 7x^{2} – 14x – 26 = 0 
n) 8a – 32 – a^{2} + 3a = a^{2} – 9a + 18 
8) Solve these systems of equations. Check your answers.
a)
x – y = 2 becomes (6 – y) – y = 2 6 – 2y = 2 so y = 2 so x = 4 
b)
2x + y = 10 becomes 2x + (2x – 2) = 10 4x = 12 so x = 3 and y = 4 
c)
3x – y = 5 becomes 3(11 – 2y) – y = 5 – 7y = – 28 so y = 4 so x = 3 
d)
3x = y + 1 becomes 3x = 9 – 2x + 1 5x = 10 so x = 2 and y = 5 
e)
3x – 2y = 13 becomes
13y^{2} + 52y – 65 = 0 y^{2} + 4y – 5 = 0 (y + 5)(y – 1) = 0 so, y = 5 or y = 1

f)
– 3x + y = 8 becomes y = 3x + 8 x^{2} + (3x + 8)^{2} = 8 x^{2} + 9x^{2} + 48x + 64 = 8 10x^{2} + 48x + 56 = 0 5x^{2} + 24x + 28 = 0 (5x + 14)(x + 2) = 0 so x = – 2 or

g)
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

h) y = x^{2} – 3x – 4

9) Solve these inequalities.
a) 3x – 2 < x + 6
so x < 4 
b) 4x + 3 > 5x – 17
so x < 20 
c)  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) x^{2} – 4x < 5 x^{2} – 4x – 5 < 0 (x – 5)(x + 1) < 0 so – 1 < x < 5 
f) 2x^{2} > 5x + 3 2x^{2} – 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:
– 3 < x < – 2 or 1/3 < x < 7 
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