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

2 min read Jun 16, 2024

Simplifying the Expression: ((1/x)-1)/(x-1)

This article will walk you through the steps of simplifying the expression ((1/x)-1)/(x-1). We will use algebraic manipulation to rewrite the expression in a simpler form.

Step 1: Simplifying the Numerator

Let's start by simplifying the numerator, (1/x)-1. To do this, we need a common denominator:

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

Step 2: Rewriting the Expression

Now our expression becomes:

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

Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite this as:

((1-x)/x) * (1/(x-1))

Step 3: Simplifying the Expression

Finally, we can multiply the numerators and denominators:

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

Notice that the numerator and denominator share a common factor of (1-x). We can cancel this factor, leaving us with:

-1/x

Conclusion

Therefore, the simplified form of the expression ((1/x)-1)/(x-1) is -1/x. This simplification process demonstrates how to manipulate fractions and common factors to arrive at a simpler expression.