Working with integers one really one
needs to talk about adding (positive
or negative numbers)
2.4 – Adding Rational Expressions
The same rules apply as when adding (or subtracting) any rational numbers (i.e. fractions).
a)
b)
c)
d)
e)
f)
g)
Factor first
Determine the common denominator
Create equivalent expressions (look for what each term is missing)
Add numerators (be careful as you may need to distribute a negative?)
Simplify resulting expression
State restrictions
Optional. Can use test value(s) (i.e. “1”) to check original and simplified expression are
truly equivalent. Both equations should return the same result.
Example 1:
a)
Writing the
numerators over
the common
denominator reenforces the
correct thinking.
We start with some simple fractions to review basic techniques involved in
adding rational numbers. Simplify the following expression.
Example 2:
a)
(x + 2) is the
common
denominator.
Watch negative.
Use working
brackets to
avoid mistakes
Trying
substituting
“1” in both
expressions to
confirm they
are the same.
b)
2 3
+
7 7
2+3
=
7
5
=
7
3 2
−
5 3
9 10
=
−
15 15
−1
=
15
3 2
−
5 3
(3)3 (5) 2
=
−
(3)5 (5)3
9 − 10
=
15
−1
=
15
Same questions done,
showing the multipliers
used to create the
equivalent expressions.
Following this technique
as the questions become
more complicated helps
avoid mistakes.
Simplify the following expression. State all restrictions
3x − 2 x + 1
−
x+2 x+2
(3 x − 2) − ( x + 1)
=
x+2
3x − 2 − x − 1
=
x+2
2x − 3
, x ≠ −2
=
x+2
b)
=
=
=
=
d)
c)
3
2
−
5
e)
=
=
=
=
2.4 – adding rational expressions
c)
=
=
=
=
=
y
3
f)
−
+1
2 y − 4 3y − 6
y
3
1
−
+
2( y − 2) 3( y − 2) 1
(3) y
( 2 )3
6 ( y − 2)
−
+
(3)( y − 2) ( 2)3( y − 2) 6( y − 2)
3 y − 6 + 6 y − 12
6( y − 2 )
Combine steps
to save time and
9 ( y − 2)
space. But be
6( y − 2 )
careful!
3
, y≠2
2
Looks close
but must be
identical so
try to rearrange.
3
5
+
x−2 2− x
5
3
+
x−2 −x+2
5
3
+
x − 2 − ( x − 2)
5
3
−
x − 2 ( x − 2)
2
, x≠2
x−2
Watch
constant
terms!
3
4x
3x
(3 x )3
( 4 )5
=
−
2
(3 x ) 4 x
( 4)3 x 3
9 x − 20
=
, x≠0
12 x 3
2x −1 x + 3
+
3
2
( 2)( 2 x − 1) (3)( x + 3)
+
( 2)3
(3) 2
4 x − 2 + 3x + 9
6
6 is the common
7x + 7
denominator.
6
Multiply terms
accordingly.
7( x + 1)
6
3
=
=
=
5
x + 5 x + 6 x − x − 12
3
5
−
( x + 2)( x + 3) ( x − 4)( x + 3)
3( x − 4) − 5( x + 2)
( x + 2)( x − 4)( x + 3)
− 2 x − 22
( x + 2)( x − 4)( x + 3)
− 2( x + 11)
, x ≠ −3 − 2,4
( x + 2)( x − 4)( x + 3)
2
=
−
Negative
denominator?
Move negative
up top.
2
2.4 – Adding Rational Expressions Practice Questions
1. Simplify the following expressions and state any restrictions.
a)
2
5
−
x +1 x +1
b)
2
+4
x2
c)
3x 5 x x
−
+
8
6 3
d)
2x + 3y 4x − y
−
5
2
e)
2
3
+
x−2 2− x
f)
2
3
+
x +1 x + 2
g)
3
1
−
3 x + 15 6 x + 24
h)
2x
3
+
x − 4x − 5 x − 5
i)
6x
3
− 2
6 x − 6 3x − 4 x + 1
j)
4y
2y
+ 2
y − 9 y + 18 y − 11y + 30
2
2. Simplify;
2
k)
2x −1
2x + 1
+ 2
2 x + 3x + 1 3x + 4 x + 1
2
2 x + 5 x − 5 3x
×
+
x − 5 x +1 x +1
3. Two triangles have the same base length of x. One has a height of x -1 while the other has a
height of x + 3. Write a simplified expression for the total area of both triangles.
−16 x + 11 y
−3
−1
x
2 + 4x 2
, x ≠ −1 b)
d)
e)
,x ≠ 2
, x ≠ 0 c) −
8
10
x +1
x−2
x2
5x + 7
5 x + 19
5x + 3
, x ≠ −1,−2 g)
, x ≠ −4,−5 h)
, x ≠ −1,5
f)
6( x + 5)( x + 4)
( x − 5)( x + 1)
( x + 1)( x + 2)
Answers 1. a)
i)
2 y (3 y − 13)
3x 2 − x + 3
1
, y ≠ 3,5,6
, x ≠ ,1 j)
( y − 3)( y − 6)( y − 5)
( x − 1)(3 x − 1)
3
k)
10 x 2 + 3 x
1 1
, x ≠ −1,− ,−
( 2 x + 1)( x + 1)(3 x + 1)
2 3
2.4 – adding rational expressions
2. a) 5, x≠-1,5 3. At=x(x+1)