Unit 3: Making Sense of Rational Expressions 173
Practice
Factor each of these and then simplify. Look for hints within the problem. Refer
to the previous page as necessary. Show essential steps.
1.
2.
Sometimes, it is necessary to factor both the numerator and denominator.
Examine the example below, then simplify each of the following expressions.
Example:
Note: The x’s do
not cancel.
3.
4.
a2 – 3a + 2 = a – 2
b2 – 2b – 3 = b – 3
x2 – 4 = x2 + x – 6
(x + 2)(x – 2)
(x + 3)(x – 2) = (x + 2)
(x + 3)
1
1
= x + 2
x + 3
2r
2 + r – 6 = r
2 + r – 2
x2 + x – 2 = x2 – 1
174 Unit 3: Making Sense of Rational Expressions
Practice
Simplify each expression. Show essential steps.
1.
2.
3.
4.
5b – 10 = b – 2
6a – 9 = 10a – 15
9x + 3 = 9
6b + 9 = 12
Unit 3: Making Sense of Rational Expressions 175
5.
6.
7.
8.
3a2
b + 6ab – 9b2
= 3b
x2 – 16 = x + 4
2a – b = b2 – 4a2
6×2 + 2 = 9×2 + 3
176 Unit 3: Making Sense of Rational Expressions
Practice
Factor each of these expressions and then simplify. Show essential steps.
1.
2.
3.
4.
y2
+ 5y – 14 = y + 2
a2
– 5a + 4 = a – 4
6m2
– m – 1 = 2m2
+ 9m – 5
2×2
+ x – 6 = 4×2 – 9
Unit 3: Making Sense of Rational Expressions 177
Practice
Use the list below to write the correct term for each definition on the line provided.
denominator
expression
fraction
rational expression
real numbers
variable
________________________ 1. a collection of numbers, symbols,
and/or operation signs that stands for a
number
________________________ 2. the top number of a fraction, indicating
the number of equal parts being
considered
________________________ 3. the bottom number of a fraction,
indicating the number of equal parts a
whole was divided into
________________________ 4. the set of all rational and irrational
numbers
________________________ 5. any part of a whole
________________________ 6. a fraction whose numerator and/or
denominator are polynomials
________________________ 7. any symbol, usually a letter, which
could represent a number
________________________ 8. a monomial or sum of monomials; any
rational expression with no variable in
the denominator
________________________ 9. the result of dividing two numbers
numerator
polynomial
quotient
178 Unit 3: Making Sense of Rational Expressions
Practice
Use the list below to complete the following statements.
canceling
cross multiplication
equivalent
factor
integers
product
simplify an expression
terms
1. If you multiply both the numerator and the denominator by the same
number, the new fraction will be because
it is the same number expressed in a different form.
2. are the numbers in the set
{… , -4, -3, -2, -1, 0, 1, 2, 3, 4, …}.
3. If you divide a numerator and a denominator by a common factor to
write a fraction in lowest terms or before multiplying fractions, you
are .
4. To , you need to perform as many of the
indicated operations as possible.
5. Numbers, variables, products, or quotients in an expression are
called .
Unit 3: Making Sense of Rational Expressions 179
6. A is a number or expression that divides
evenly into another number.
7. When you multiply numbers together, the result is called the
.
8. To find a missing numerator or denominator in equivalent fractions
or ratios, you can use a method called
and make the cross products equal.
206 Unit 3: Making Sense of Rational Expressions
Practice
Solve and check each equation. Use the examples on pages 200-205 for
reference. Show essential steps.
Hint: Find a step that looks similar to the problem you need help with and
follow from that point.
Remember: To check your work, replace the variable in the
original problem with the answer you found.
1. 3x – 7 = 17
2. 4x + 20 = x – 4
3. x
6 = 1.5
4. 2x
5 = 3.2
Unit 3: Making Sense of Rational Expressions 207
5. 5(x – 4) = 20
6. 5(4x – 7) = 0
7. 8x – 2x = 42
8. 5x – 3 = 2x + 18
9. -2x + 4 = -4x – 10
208 Unit 3: Making Sense of Rational Expressions
Practice
Solve and check each equation. Use the examples on pages 200-205 for
reference. Show essential steps.
1. 2(3x – 4) + 6 = 10
2. 3(x – 7) – x = -9
3. 2
3 x = 1
Hint: 2
3 x =
2x
3 . Rewrite 1 as 1
1 and cross multiply.
4. -1
2 x – 3
4 = 4
Unit 3: Making Sense of Rational Expressions 209
5. -3x = -33
8
6. -2
x = 8
7. -3x – 3
2 = 11
2
210 Unit 3: Making Sense of Rational Expressions
Practice
Solve and check each equation.
1. -87 = 9 – 8x
2. 4k + 3 = 3k + 1
3. 5a + 9 = 64
4. b
3 + 5 = -2
Unit 3: Making Sense of Rational Expressions 211
5. 4x = -(9 – x)
6. x
5 = -10
7. 3x – 1 = -x + 19
212 Unit 3: Making Sense of Rational Expressions
Practice
Solve and check each equation. Reduce fractions to simplest form.
1. 5x – 3 = 2x + 18
2. 6x – (4x – 12) = 3x + 5
3. x
6 = -24
5
4. 4(x – 2) = -3(x + 5)
Unit 3: Making Sense of Rational Expressions 213
5. 5( 1
3 x – 2) = 4
6. x
4 + 3
2 = 5
8
7. 9
2
x = 5
1
8. -1
2 + 8x
5 = -7
8

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