Problem 8 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To build \(\frac{8 x}{2-x}\) so that it has a denominator of \(x-2,\) we multiply it by 1 in the form of _____.

Step-by-Step Solution

Answer

Multiply by \((-1)\cdot\frac{x-2}{x-2}\).

Step 1: Identify the Required Denominator

The problem states that we need to change the denominator from \(2-x\) to \(x-2\). Notice that \(x-2 = -(2-x)\). This means, we simply need to multiply by \(-1\) to reverse the sign in the denominator.

Step 2: Determine the Correct Form of 1

We want to multiply by 1 in a form that changes the sign of the denominator. The expression we use should look like \((x-2)/(x-2)\) so that the entire expression is multiplied by 1, which doesn't change its value but changes the sign of the denominator.

Step 3: Apply the Multiplication

Multiply \(\frac{8x}{2-x}\) by \(-1\) in the form \((-1)/(-1)\) or equivalently \((-1)\cdot\frac{x-2}{x-2}\). This results in \(-\frac{8x}{x-2}\). The denominator is now \(x-2\), as needed.

Key Concepts

Denominator Transformation

Denominator transformation is a concept that involves changing the denominator of a fraction to another without altering the value of the entire expression. This may seem complex at first, but it's all about smart manipulation of mathematical expressions. In algebra, denominators are often rearranged to simplify an expression or to make it compatible with another fraction for operations like addition or subtraction.

To understand this, let's consider the transformation from the denominator of \( \frac{8x}{2-x} \) to \( x-2 \). At first glance, these seem different, but they are related through multiplication by \(-1\). Note that:

• \(x-2 = -(2-x)\)
• This implies multiplying the entire expression by \(-1\)

By transforming \( 2-x \) into \( x-2 \), we've made it easier to work with in algebraic fractions without altering the expression's value.

Multiplying by One

Multiplying by one is a mathematical trick used to alter expressions while keeping them equivalent in value. In algebra, this principle comes in incredibly handy. The beauty of one is that you can multiply any number or expression by it, and it remains unchanged in value.

When dealing with altering denominators, multiplying by one in a clever form like \( \frac{x-2}{x-2} \) is crucial. This step allows us to change expressions without impacting their content. Take the exercise's solution as an example:

• Instead of changing the number's value, we change its form.
• By multiplying \( \frac{8x}{2-x} \) by \( \frac{x-2}{x-2} \), you effectively invert the denominator's sign, turning \( 2-x \) into \( x-2 \).

This trick ensures the expression remains equivalent to its original form, merely adjusted for compatibility with other algebraic operations.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Understanding how these expressions work is foundational for algebra. Manipulating rational expressions involves operations like factorization, simplification, and finding common denominators.

Dealing with rational expressions often requires rewriting denominators, as seen in the example of \( \frac{8x}{2-x} \). Through thoughtful use of algebraic properties, like the transformation of the denominator, we achieve a more convenient form, such as \( \frac{8x}{x-2} \).

In working with these structures, always remember:

• The goal is to simplify or solve the equation while keeping it equivalent.
• In transformations, multiplication by a form of one is used to adjust without changing value.

Rational expressions challenge you to think strategically about rearranging parts to solve a problem, offering a satisfying puzzle-solving experience.