Rational Equations
and Inequalities

Martin-Gay,
Developmental Mathematics
2
14.1 – Simplifying Rational Expressions
14.2 – Multiplying and Dividing Rational Expressions
14.3 – Adding and Subtracting Rational Expressions with
the Same Denominator and Least Common Denominators
14.4 – Adding and Subtracting Rational Expressions with Dif
ferent Denominators
14.5 – Solving Equations Containing Rational Expressions
14.6 – Problem Solving with Rational Expressions
14.7 – Simplifying Complex Fractions
Chapter Sections

Simplifying Rational
Expressions

Martin-Gay,
Developmental Mathematics
4
Simplifying Rational Expressions
RULE:
1.Factor the numerator and denominator.
2.Write a product of two rational expressions,
one factor containing the GCF of the numerator
and denominator, and the other containing
the
remaining factors.
3.Rewrite the factor containing the GCF as 1.
4.Multiply the remaining factors by 1.

Martin-Gay,
Developmental Mathematics
5
Rational Expressions
Q
P
Rational expressions
can be written in the form
where
P
and
Q
are both polynomials and
Q
0.
It is also called
Algebraic Fractions.
Examples of Rational Expressions
5
4
4
2
3
2
x
x
x
2
2
4
3
2
3
4
y
xy
x
y
x
4
3
2
x

Martin-Gay,
Developmental Mathematics
6
Rational Expressions
The following are NOT rational expressions
x
x
5
2
2
2
1
x
x
1
4
2
3
x
x

Martin-Gay,
Developmental Mathematics
7
To evaluate a rational expression for a particular
value(s), substitute the replacement value(s) into the
rational expression and simplify the result.
Evaluating Rational Expressions
Example
Evaluate the following expression for y =
2.
y
y
5
2
)
2
2
(
2
5
7
4
7
4

Martin-Gay,
Developmental Mathematics
8
In the previous example, what would happen if we
tried to evaluate the rational expression for y = 5?
y
y
5
2
5
2
5
5
0
3
This expression is undefined!
Evaluating Rational Expressions

Martin-Gay,
Developmental Mathematics
9
We have to be able to determine when a
rational expression is undefined.
A rational expression is undefined when the
denominator is equal to zero.
The numerator being equal to zero is okay
(the rational expression simply equals zero).
Undefined Rational Expressions

Martin-Gay,
Developmental Mathematics
10
Find any real numbers that make the following rational
expression undefined.
45
15
4
9
3
x
x
x
The expression is undefined when 15
x
+ 45 = 0.
So the expression is undefined when
x
=
3.
Undefined Rational Expressions
Example

Martin-Gay,
Developmental Mathematics
11
Simplifying
a rational expression means writing it in
lowest terms or simplest form.
To do this, we need to use the
Fundamental Principle of Rational Expressions
If
P
,
Q
, and
R
are polynomials, and
Q
and
R
are not 0,
Q
P
QR
PR
Simplifying Rational Expressions

Martin-Gay,
Developmental Mathematics
12
Simplifying a Rational Expression
1)
Completely factor the numerator and
denominator.

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- Fall '16
- Sherwin C. Lagrama
- Fractions, Fraction, Elementary arithmetic, Rational function