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Simplify (14/(x-9)+x/(x-6))/(3/(x-10)-2/(x-9))
14
x
-
9
+
x
x
-
6
3
x
-
10
-
2
x
-
9
Step 1
Multiply
the
numerator
and
denominator
of the
fraction
by
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
Multiply
14
x
-
9
+
x
x
-
6
3
x
-
10
-
2
x
-
9
by
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
⋅
14
x
-
9
+
x
x
-
6
3
x
-
10
-
2
x
-
9
Combine
.
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
14
x
-
9
+
x
x
-
6
)
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
3
x
-
10
-
2
x
-
9
)
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
14
x
-
9
+
x
x
-
6
)
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
3
x
-
10
-
2
x
-
9
)
Step 2
Apply the
distributive property
.
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
14
x
-
9
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Step 3
Simplify by cancelling.
Cancel the
common factor
of
x
-
9
.
Factor
x
-
9
out of
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
(
x
-
9
)
(
(
x
-
6
)
(
x
-
10
)
)
14
x
-
9
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
.
(
x
-
9
)
(
(
x
-
6
)
(
x
-
10
)
)
14
x
-
9
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Rewrite the
expression
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
of
x
-
6
.
Factor
x
-
6
out of
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
6
)
(
(
x
-
9
)
(
x
-
10
)
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
6
)
(
(
x
-
9
)
(
x
-
10
)
)
x
x
-
6
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Rewrite the
expression
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
of
x
-
10
.
Factor
x
-
10
out of
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
10
)
(
(
x
-
9
)
(
x
-
6
)
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
10
)
(
(
x
-
9
)
(
x
-
6
)
)
3
x
-
10
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Rewrite the
expression
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
(
-
2
x
-
9
)
Cancel the
common factor
of
x
-
9
.
Move the leading negative in
-
2
x
-
9
into the
numerator
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
-
2
x
-
9
Factor
x
-
9
out of
(
x
-
9
)
(
x
-
6
)
(
x
-
10
)
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
9
)
(
(
x
-
6
)
(
x
-
10
)
)
-
2
x
-
9
Cancel the
common factor
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
9
)
(
(
x
-
6
)
(
x
-
10
)
)
-
2
x
-
9
Rewrite the
expression
.
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Step 4
Simplify the
numerator
.
Factor
x
-
10
out of
(
x
-
6
)
(
x
-
10
)
⋅
14
+
(
x
-
9
)
(
x
-
10
)
x
.
Factor
x
-
10
out of
(
x
-
6
)
(
x
-
10
)
⋅
14
.
(
x
-
10
)
(
(
x
-
6
)
⋅
14
)
+
(
x
-
9
)
(
x
-
10
)
x
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Factor
x
-
10
out of
(
x
-
9
)
(
x
-
10
)
x
.
(
x
-
10
)
(
(
x
-
6
)
⋅
14
)
+
(
x
-
10
)
(
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Factor
x
-
10
out of
(
x
-
10
)
(
(
x
-
6
)
⋅
14
)
+
(
x
-
10
)
(
(
x
-
9
)
x
)
.
(
x
-
10
)
(
(
x
-
6
)
⋅
14
+
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
(
x
-
10
)
(
(
x
-
6
)
⋅
14
+
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Apply the
distributive property
.
(
x
-
10
)
(
x
⋅
14
-
6
⋅
14
+
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Move
14
to the left of
x
.
(
x
-
10
)
(
14
⋅
x
-
6
⋅
14
+
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Multiply
-
6
by
14
.
(
x
-
10
)
(
14
⋅
x
-
84
+
(
x
-
9
)
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Apply the
distributive property
.
(
x
-
10
)
(
14
x
-
84
+
x
⋅
x
-
9
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Multiply
x
by
x
.
(
x
-
10
)
(
14
x
-
84
+
x
2
-
9
x
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Subtract
9
x
from
14
x
.
(
x
-
10
)
(
5
x
-
84
+
x
2
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Reorder
terms
.
(
x
-
10
)
(
x
2
+
5
x
-
84
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Factor
x
2
+
5
x
-
84
using the AC method.
Consider the form
x
2
+
b
x
+
c
. Find a pair of
integers
whose
product
is
c
and whose
sum
is
b
. In this case, whose
product
is
-
84
and whose
sum
is
5
.
-
7
,
12
Write the factored form using these
integers
.
(
x
-
10
)
(
(
x
-
7
)
(
x
+
12
)
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Step 5
Simplify the
denominator
.
Factor
x
-
6
out of
(
x
-
9
)
(
x
-
6
)
⋅
3
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
.
Factor
x
-
6
out of
(
x
-
9
)
(
x
-
6
)
⋅
3
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
(
x
-
9
)
⋅
3
)
+
(
x
-
6
)
(
x
-
10
)
⋅
-
2
Factor
x
-
6
out of
(
x
-
6
)
(
x
-
10
)
⋅
-
2
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
(
x
-
9
)
⋅
3
)
+
(
x
-
6
)
(
(
x
-
10
)
⋅
-
2
)
Factor
x
-
6
out of
(
x
-
6
)
(
(
x
-
9
)
⋅
3
)
+
(
x
-
6
)
(
(
x
-
10
)
⋅
-
2
)
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
(
x
-
9
)
⋅
3
+
(
x
-
10
)
⋅
-
2
)
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
(
x
-
9
)
⋅
3
+
(
x
-
10
)
⋅
-
2
)
Apply the
distributive property
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
x
⋅
3
-
9
⋅
3
+
(
x
-
10
)
⋅
-
2
)
Move
3
to the left of
x
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
3
⋅
x
-
9
⋅
3
+
(
x
-
10
)
⋅
-
2
)
Multiply
-
9
by
3
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
3
⋅
x
-
27
+
(
x
-
10
)
⋅
-
2
)
Apply the
distributive property
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
3
x
-
27
+
x
⋅
-
2
-
10
⋅
-
2
)
Move
-
2
to the left of
x
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
3
x
-
27
-
2
⋅
x
-
10
⋅
-
2
)
Multiply
-
10
by
-
2
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
3
x
-
27
-
2
⋅
x
+
20
)
Subtract
2
x
from
3
x
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
x
-
27
+
20
)
Add
-
27
and
20
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
x
-
7
)
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
x
-
7
)
Step 6
Cancel the
common factor
of
x
-
7
.
Cancel the
common factor
.
(
x
-
10
)
(
x
-
7
)
(
x
+
12
)
(
x
-
6
)
(
x
-
7
)
Rewrite the
expression
.
(
x
-
10
)
(
x
+
12
)
x
-
6
(
x
-
10
)
(
x
+
12
)
x
-
6
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