Question

Simplify: \(\frac{\frac{1}{x}+1}{\frac{1}{x^{2}}-1}\)

Short answer

\( \frac{x}{1-x} \)

Step-by-step solution

Simplify the numerator

The numerator is given as \(\frac{1}{x} + 1\). Let's rewrite this as \[ \frac{1}{x} + 1 = \frac{1}{x} + \frac{x}{x} = \frac{1 + x}{x} \].

Simplify the denominator

The denominator is given as \(\frac{1}{x^2} - 1\). We can rewrite this as \[ \frac{1}{x^2} - 1 = \frac{1 - x^2}{x^2} \].

Combine simplified numerator and denominator

Now, we have the simplified fraction \[ \frac{\frac{1 + x}{x}}{\frac{1 - x^2}{x^2}} \].

Divide the fractions

Dividing the two fractions is the same as multiplying by the reciprocal of the denominator: \[ \frac{\frac{1 + x}{x}}{\frac{1 - x^2}{x^2}} = \frac{1 + x}{x} \times \frac{x^2}{1 - x^2} \].

Simplify the expression

Simplify the expression by multiplying: \(\frac{1 + x}{x} \times \frac{x^2}{1 - x^2} = \frac{(1 + x) \cdot x^2}{x \cdot (1 - x^2)}\). Cancel out the common terms: \[ = \frac{(1 + x) \cdot x}{1 - x^2} \].

Factor the denominator

Notice that the denominator \(1 - x^2\) can be factored using the difference of squares: \[ 1 - x^2 = (1 - x)(1 + x) \].

Further simplify

Now, substituting the factored form of the denominator, we get: \[ \frac{(1 + x) \cdot x}{(1 - x)(1 + x)} \]. Cancel out the common \(1 + x\) term: \[ = \frac{x}{1 - x} \].

Key concepts

Algebraic Manipulation

To simplify a rational expression like the one in this exercise, algebraic manipulation is essential. This skill involves rearranging and combining terms using basic algebraic rules.

Start by rewriting complex fractions. In our case, the numerator \(\frac{1}{x} + 1\) and denominator \(\frac{1}{x^2} - 1\) need to be simplified.

For the numerator, we use a common denominator: \[ \frac{1}{x} + 1 = \frac{1 + x}{x} \].

In the denominator, we also find a common denominator: \[ \frac{1}{x^2} - 1 = \frac{1 - x^2}{x^2} \].

Another key step in algebraic manipulation is understanding how to work with reciprocals. When dividing fractions, you can multiply by the reciprocal of the divisor. In this case: \[ \frac{\frac{1 + x}{x}}{\frac{1 - x^2}{x^2}} = \frac{1 + x}{x} \times \frac{x^2}{1 - x^2} \].

Mastering algebraic manipulation helps simplify expressions and solve equations more effectively.

Fractions

Fractions are a critical part of algebra and understanding them is key to simplifying expressions.

In this problem, we are dealing with complex fractions, which means fractions within fractions. To simplify, look for common terms and apply basic fraction rules.

For the numerator:

• Combine terms with a common denominator: \[ \frac{1}{x} + 1 = \frac{1}{x} + \frac{x}{x} = \frac{1 + x}{x} \]

For the denominator:

• Also combine terms with a common denominator: \[ \frac{1}{x^2} - 1 = \frac{1 - x^2}{x^2} \]

Next, separate the fractions and multiply by the reciprocal: \[ \frac{\frac{1 + x}{x}}{\frac{1 - x^2}{x^2}} = \frac{1 + x}{x} \times \frac{x^2}{1 - x^2} \].

This approach simplifies the complex fraction into a more manageable form.

Remember, understanding fractions involves recognizing common terms and knowing how to manipulate the numerator and the denominator effectively.

Factoring Polynomials

Factoring polynomials is key for simplifying rational expressions. It breaks down complex expressions into simpler, multipliable parts.

In this exercise, the denominator \(1 - x^2\) is a perfect example. It can be factored using the difference of squares rule:

• Recognize the form: \(a^2 - b^2 = (a - b)(a + b)\)
• Substitute: \(1 - x^2 = (1 - x)(1 + x)\)

Now we have: \[ \frac{(1 + x) \cdot x}{(1 - x)(1 + x)} \], and can cancel common terms.

This leaves: \[ \frac{x}{1 - x} \].

Factoring polynomials helps in reducing expressions and finding solutions. It is especially useful when dealing with higher-degree polynomials or when simplifying complex rational expressions.