To simplify the expression 2 / (x² + x) – 1 / x, we first need to find a common denominator. The terms in the expression are:
- 2 / (x² + x)
- -1 / x
The first term has a denominator of x² + x, which can be factored as x(x + 1). The second term has a denominator of x. To combine these fractions, our common denominator will be x(x + 1).
Now, we can rewrite the fractions:
- 2 / (x² + x) = 2 / [x(x + 1)]
- -1 / x = -1(x + 1) / [x(x + 1)] = – (x + 1) / [x(x + 1)]
Now we can combine the fractions over the common denominator:
(2 – (x + 1)) / [x(x + 1)]
Now we need to simplify the numerator:
2 – (x + 1) = 2 – x – 1 = 1 – x
So our expression now looks like:
(1 – x) / [x(x + 1)]
This is the simplified form of the original expression. Thus, the final answer is:
(1 – x) / [x(x + 1)]