# Add -5/(x-1)+(2-x)/x

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In order to add fractions, they must have common denominators. The common denominator in this case is the product of the two given denominators. Therefore, the common denominator is x(x - 1).

Multiply the first fraction by x.

-5 / (x - 1) = -5x / x(x - 1)

Multiply the second fraction by (x - 1)

(2 - x)(x - 1) / x(x - 1)

Now that the fractions have common denominators, you are able to add their numerators.

numerator: -5x + (2 - x)(x - 1)

denominator: x(x - 1)

Multiply the binomials in the numerator using FOIL.

-5x + 2x - 2 - x^2 + x

Combine like terms.

-x^2 - 2x - 2

Use the distributive property in the denominator.

x(x - 1)

x^2 - x

To avoid negative coefficients, multiply everything by -1.

numerator: -1(-x^2 - 2x - 2) = x^2 + 2x + 2

denominator: -1(x^2 - x) = -x^2 + x = x - x^2

Simplify the denominator by factoring out the x.

x - x^2 = x(1 - x)

So now you have this:

numerator: x^2 + 2x + 2

denominator: x(1 - x)

**Simplified answer: (x^2 + 2x + 2) / [x(1 - x)]**

I see I made a small mistake above in multiplying the binomials (2-x)(x-1). justaguide is correct. Here is my corrected solution:

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I believe you mean:

-5/(x-1) + (2-x)/x

In other words, the x-1 is the bottom of the first fraction and the 2-x is the top of the second fraction. In this case, we simplify as follows:

First, we find a common denominator. It is x(x-1).

Now we must multiply the first fractions numerator by (x) and the second fraction by (x-1) in order to have the common denominator.

Now we have:

-5x/[x(x-1)] + [(2-x)(x-1)]/[x(x-1)]

= -5x/[x(x-1)] + [2x-2-x^2+x][x(x-1)]

= -5x/[x(x-1)] + [-x^2+3x-2][x(x-1)]

Now combining the numerators:

= [-5x-x^2+3x-2][x(x-1)]

= [-x^2-2x-2][x(x-1)]

= -[x^2+2x+2][x(x-1)]

**= -(x^2+2x+2)[x(1-x)]**

We have to simplify: -5/(x-1)+(2-x)/x

-5/(x-1)+(2-x)/x

=> -5x/x(x-1) + (2 - x)(x - 1)/x(x-1)

=> [-5x + (2 - x)(x - 1)]/x(x-1)

=> [-5x + 2x - 2 - x^2 + x]/x(x-1)

=> [-2x - 2 - x^2]/x(x-1)

=> [-2x - 2 - x^2]/x(x-1)

=> (x^2 + 2x + 2)/x(1 - x)

**The simplified form is : (x^2 + 2x + 2)/x(1 - x)**

I believe you mean:

-5/(x-1) + (2-x)/x

In other words, the x-1 is the bottom of the first fraction and the 2-x is the top of the second fraction. In this case, we simplify as follows:

First, we find a common denominator. It is x(x-1).

Now we must multiply the first fractions numerator by (x) and the second fraction by (x-1) in order to have the common denominator.

Now we have:

-5x/[x(x-1)] + [(2-x)(x-1)]/[x(x-1)]

= -5x/[x(x-1)] + [2x-2x-x^2+x][x(x-1)]

= -5x/[x(x-1)] + [-x^2+x][x(x-1)]

Now combining the numerators:

= [-5x-x^2+x][x(x-1)]

= [-4x-x^2][x(x-1)]

= [x(-4-x)][x(x-1)]

Cancelling the x's (one on top and one on the bottom), we have:

**= (-x-4)/(x-1) **

**or if you want to factor out -1 on top:**

**= -(x+4)/(x-1)**

X^2-2X-2/X^2-X