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Simplify (14/(x-9)+x/(x-6))/(3/(x-10)-2/(x-9))

14x-9+xx-63x-10-2x-9

Step 1
Multiply the numerator and denominator of the fraction by (x-9)(x-6)(x-10).

Multiply 14x-9+xx-63x-10-2x-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)14x-9+xx-63x-10-2x-9
Combine.
(x-9)(x-6)(x-10)(14x-9+xx-6)(x-9)(x-6)(x-10)(3x-10-2x-9)
(x-9)(x-6)(x-10)(14x-9+xx-6)(x-9)(x-6)(x-10)(3x-10-2x-9)

Step 2
Apply the distributive property.
(x-9)(x-6)(x-10)14x-9+(x-9)(x-6)(x-10)xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-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))14x-9+(x-9)(x-6)(x-10)xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
Cancel the common factor.
(x-9)((x-6)(x-10))14x-9+(x-9)(x-6)(x-10)xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
Rewrite the expression.
(x-6)(x-10)14+(x-9)(x-6)(x-10)xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
(x-6)(x-10)14+(x-9)(x-6)(x-10)xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-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))xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
Cancel the common factor.
(x-6)(x-10)14+(x-6)((x-9)(x-10))xx-6(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
Rewrite the expression.
(x-6)(x-10)14+(x-9)(x-10)x(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-9)
(x-6)(x-10)14+(x-9)(x-10)x(x-9)(x-6)(x-10)3x-10+(x-9)(x-6)(x-10)(-2x-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))3x-10+(x-9)(x-6)(x-10)(-2x-9)
Cancel the common factor.
(x-6)(x-10)14+(x-9)(x-10)x(x-10)((x-9)(x-6))3x-10+(x-9)(x-6)(x-10)(-2x-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)(-2x-9)
(x-6)(x-10)14+(x-9)(x-10)x(x-9)(x-6)3+(x-9)(x-6)(x-10)(-2x-9)
Cancel the common factor of x-9.

Move the leading negative in -2x-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)-2x-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))-2x-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))-2x-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)(x14-614+(x-9)x)(x-9)(x-6)3+(x-6)(x-10)-2
Move 14 to the left of x.
(x-10)(14x-614+(x-9)x)(x-9)(x-6)3+(x-6)(x-10)-2
Multiply -6 by 14.
(x-10)(14x-84+(x-9)x)(x-9)(x-6)3+(x-6)(x-10)-2
Apply the distributive property.
(x-10)(14x-84+xx-9x)(x-9)(x-6)3+(x-6)(x-10)-2
Multiply x by x.
(x-10)(14x-84+x2-9x)(x-9)(x-6)3+(x-6)(x-10)-2
Subtract 9x from 14x.
(x-10)(5x-84+x2)(x-9)(x-6)3+(x-6)(x-10)-2
Reorder terms.
(x-10)(x2+5x-84)(x-9)(x-6)3+(x-6)(x-10)-2
Factor x2+5x-84 using the AC method.

Consider the form x2+bx+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)(x3-93+(x-10)-2)
Move 3 to the left of x.
(x-10)(x-7)(x+12)(x-6)(3x-93+(x-10)-2)
Multiply -9 by 3.
(x-10)(x-7)(x+12)(x-6)(3x-27+(x-10)-2)
Apply the distributive property.
(x-10)(x-7)(x+12)(x-6)(3x-27+x-2-10-2)
Move -2 to the left of x.
(x-10)(x-7)(x+12)(x-6)(3x-27-2x-10-2)
Multiply -10 by -2.
(x-10)(x-7)(x+12)(x-6)(3x-27-2x+20)
Subtract 2x from 3x.
(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