# Simplify ((y+2)/(y-1)-(y-3)/(y-2))/(y+2)

y+2y-1-y-3y-2y+2
Multiply the numerator and denominator of the complex fraction by (y-1)(y-2).
Multiply y+2y-1-y-3y-2y+2 by (y-1)(y-2)(y-1)(y-2).
(y-1)(y-2)(y-1)(y-2)⋅y+2y-1-y-3y-2y+2
Combine.
(y-1)(y-2)(y+2y-1-y-3y-2)(y-1)(y-2)(y+2)
(y-1)(y-2)(y+2y-1-y-3y-2)(y-1)(y-2)(y+2)
Apply the distributive property.
(y-1)(y-2)y+2y-1+(y-1)(y-2)(-y-3y-2)(y-1)(y-2)y+(y-1)(y-2)⋅2
Simplify by cancelling.
Cancel the common factor of y-1.
Cancel the common factor.
(y-1)(y-2)y+2y-1+(y-1)(y-2)(-y-3y-2)(y-1)(y-2)y+(y-1)(y-2)⋅2
Rewrite the expression.
(y-2)(y+2)+(y-1)(y-2)(-y-3y-2)(y-1)(y-2)y+(y-1)(y-2)⋅2
(y-2)(y+2)+(y-1)(y-2)(-y-3y-2)(y-1)(y-2)y+(y-1)(y-2)⋅2
Cancel the common factor of y-2.
Move the leading negative in -y-3y-2 into the numerator.
(y-2)(y+2)+(y-1)(y-2)-(y-3)y-2(y-1)(y-2)y+(y-1)(y-2)⋅2
Factor y-2 out of (y-1)(y-2).
(y-2)(y+2)+(y-2)(y-1)-(y-3)y-2(y-1)(y-2)y+(y-1)(y-2)⋅2
Cancel the common factor.
(y-2)(y+2)+(y-2)(y-1)-(y-3)y-2(y-1)(y-2)y+(y-1)(y-2)⋅2
Rewrite the expression.
(y-2)(y+2)+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
(y-2)(y+2)+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
(y-2)(y+2)+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Simplify the numerator.
Expand (y-2)(y+2) using the FOIL Method.
Apply the distributive property.
y(y+2)-2(y+2)+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Apply the distributive property.
y⋅y+y⋅2-2(y+2)+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Apply the distributive property.
y⋅y+y⋅2-2y-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
y⋅y+y⋅2-2y-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Combine the opposite terms in y⋅y+y⋅2-2y-2⋅2.
Reorder the factors in the terms y⋅2 and -2y.
y⋅y+2y-2y-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Subtract 2y from 2y.
y⋅y+0-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
y⋅y-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
y⋅y-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Simplify each term.
Multiply y by y.
y2-2⋅2+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply -2 by 2.
y2-4+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4+(y-1)(-(y-3))(y-1)(y-2)y+(y-1)(y-2)⋅2
Apply the distributive property.
y2-4+(y-1)(-y–3)(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply -1 by -3.
y2-4+(y-1)(-y+3)(y-1)(y-2)y+(y-1)(y-2)⋅2
Expand (y-1)(-y+3) using the FOIL Method.
Apply the distributive property.
y2-4+y(-y+3)-1(-y+3)(y-1)(y-2)y+(y-1)(y-2)⋅2
Apply the distributive property.
y2-4+y(-y)+y⋅3-1(-y+3)(y-1)(y-2)y+(y-1)(y-2)⋅2
Apply the distributive property.
y2-4+y(-y)+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4+y(-y)+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Simplify and combine like terms.
Simplify each term.
Rewrite using the commutative property of multiplication.
y2-4-y⋅y+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply y by y by adding the exponents.
Move y.
y2-4-(y⋅y)+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply y by y.
y2-4-y2+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4-y2+y⋅3-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Move 3 to the left of y.
y2-4-y2+3⋅y-1(-y)-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply -1(-y).
Multiply -1 by -1.
y2-4-y2+3y+1y-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply y by 1.
y2-4-y2+3y+y-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4-y2+3y+y-1⋅3(y-1)(y-2)y+(y-1)(y-2)⋅2
Multiply -1 by 3.
y2-4-y2+3y+y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4-y2+3y+y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4-y2+4y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
y2-4-y2+4y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
Subtract y2 from y2.
0-4+4y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
Subtract 4 from 0.
-4+4y-3(y-1)(y-2)y+(y-1)(y-2)⋅2
Subtract 3 from -4.
4y-7(y-1)(y-2)y+(y-1)(y-2)⋅2
4y-7(y-1)(y-2)y+(y-1)(y-2)⋅2
Factor (y-1)(y-2) out of (y-1)(y-2)y+(y-1)(y-2)⋅2.
Factor (y-1)(y-2) out of (y-1)(y-2)y.
4y-7(y-1)(y-2)(y)+(y-1)(y-2)⋅2
Factor (y-1)(y-2) out of (y-1)(y-2)⋅2.
4y-7(y-1)(y-2)(y)+(y-1)(y-2)(2)
Factor (y-1)(y-2) out of (y-1)(y-2)(y)+(y-1)(y-2)(2).
4y-7(y-1)(y-2)(y+2)
4y-7(y-1)(y-2)(y+2)
Simplify ((y+2)/(y-1)-(y-3)/(y-2))/(y+2)

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