To simplify the expression (8x^2 – 4x) / (4x – 2) when x is not equal to 1/2, factor out common terms:

(4x(2x – 1)) / (2(2x – 1))

Now, you can cancel out the common factor (2x – 1) in the numerator and denominator:

4x / 2

This simplifies to:

2x

## Simplifying Rational Expressions with Excluded Values

In algebra, simplifying rational expressions involves reducing fractions through factoring and canceling common terms. However, this process requires careful consideration when excluding certain disallowed values for the variables. Let’s look at simplifying the expression (8x^2 – 4x) / (4x – 2) when x cannot equal 1/2.

## Factor the Numerator

We first factor the numerator:

8x^2 – 4x = 8x(x – 1/2)

Since x ≠ 1/2, we can cancel (x – 1/2) in the numerator and denominator.

## Factor the Denominator

The denominator factors as:

4x – 2 = 2(x – 1/2)

## Cancel Common Factors

Canceling (x – 1/2) from the numerator and denominator gives:

(8x(x – 1/2)) / (2(x – 1/2)) = 8x / 2 = 4x

However, this is only valid when x ≠ 1/2. Proceeding with cancellation requires this assumption.

## Check Excluded Values

Let’s verify the simplified expression works for x = 1/2:

Original: (8(1/2)2 – 4(1/2)) / (4(1/2) – 2) = 0 / 0 = Undefined

Simplified: 4(1/2) = 2

The solutions differ, so the simplified form is invalid when x = 1/2.

Understanding how excluded values affect rational expression simplification is key. Always note any disallowed values for variables before cancelling. With this extra awareness, simplifying rationals while avoiding pitfalls becomes a straightforward process.

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