samhouston | Middle School Teacher | (Level 1) Associate Educator

Posted on

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)]

ken5000 | High School Teacher | (Level 1) Adjunct Educator

Posted on

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)]

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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)

ken5000 | High School Teacher | (Level 1) Adjunct Educator

Posted on

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)