Question

Describe and correct the error(s) made in each of the problems below.

1−x / (5−x)(−x)=x−1 / x(x−5)

5/s+2/5=2/s

Answer 1

`\displaystyle (1-x)/((5-x)(-x)) =-(x-1 )/( x(x-5))`

`\displaystyle (5)/(s)* (2)/(5) =(2)/(s)`

Explanation:

Errors in Algebraic Operations

It's usual that students make mistakes when misunderstanding the application of algebra's basic rules. Here we have two of them

• When we change the signs of all the terms of a polynomial, the expression must be preceded by a negative sign
• When multiplying negative and positive quantities, if the number of negatives is odd, the result is negative. If the number of negatives is even, the result is positive.
• Not to confuse product of fractions with the sum of fractions. Rules are quite different

The first expression is

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

Let's arrange into format:

`\displaystyle (1-x)/((5-x)(-x)) =(x-1 )/( x(x-5))`

We can clearly see in all of the factors in the expression the signs were changed correctly, but the result should have been preceeded with a negative sign, because it makes 3 (odd number) negatives, resulting in a negative expression. The correct form is

`\displaystyle (1-x)/((5-x)(-x)) =-(x-1 )/( x(x-5))`

Now for the second expression

`5/s+2/5=2/s`

Let's arrange into format

`\displaystyle (5)/(s)+(2)/(5) =(2)/(s)`

It's a clear mistake because it was asssumed a product of fractions instead of a SUM of fractions. If the result was correct, then the expression should have been

`\displaystyle (5)/(s)* (2)/(5) =(2)/(s)`