Problem 90 - Intermediate Algebra

Question

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{1}{6 x}-\frac{2}{x-6}\) b. \(\frac{1}{6 x}=\frac{2}{x-6}\)

Step-by-Step Solution

Answer

a. \(\frac{-11x - 6}{6x(x-6)}\); b. \(x = \frac{-6}{11}\).

Step 1: Find a common denominator for part (a)

The expressions given are \( \frac{1}{6x} \) and \( \frac{2}{x-6} \). Notice that the denominators 6x and x-6 are different. To combine these fractions, we need a common denominator, which would be the product \((6x)(x-6)\).

Step 2: Rewrite each fraction with the common denominator

Rewrite \( \frac{1}{6x} \) as \( \frac{(x-6)}{6x(x-6)} \) and \( \frac{2}{x-6} \) as \( \frac{12x}{6x(x-6)} \). Now the fractions have the same denominator.

Step 3: Subtract the fractions in part (a)

Subtract the numerators, keeping the common denominator: \( \frac{(x-6)-12x}{6x(x-6)} \). Simplify the numerator: \( -11x - 6 \). So, the simplified expression is \( \frac{-11x - 6}{6x(x-6)} \).

Step 4: Set up the equation for part (b)

The equation given is \( \frac{1}{6x} = \frac{2}{x-6} \). Cross-multiply to eliminate the fractions: \( 1(x-6) = 2(6x) \).

Step 5: Simplify and solve the equation

Simplify the equation from Step 4: \( x - 6 = 12x \). Subtract \( x \) from both sides to get \(-6 = 11x \). Divide by 11: \( x = \frac{-6}{11} \).

Step 6: Perform a solution check

Substitute \( x = \frac{-6}{11} \) back into the original equation to ensure equality. Both sides reduce to the same value, confirming the solution is correct.

Key Concepts

Fraction Operations

Fraction operations revolve around using common denominators to combine fractions either by addition or subtraction. This step is crucial when the denominators differ, as is common in many algebraic expressions.

To solve an expression like \( \frac{1}{6x} - \frac{2}{x-6} \), the first step is to identify a common denominator. Here's why:

• The common denominator helps align the fractions for easy subtraction or addition.
• For the expressions \( \frac{1}{6x} \) and \( \frac{2}{x-6} \), the common denominator is achieved by multiplying the two bases: \((6x)(x-6)\).
• Rewriting each fraction with this common denominator allows direct subtraction or addition of their numerators.

Once the fractions are aligned with a common denominator, you subtract or add the numerators while keeping this common denominator intact. This method simplifies the overall fraction in many algebraic operations.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. A common technique used for solving equations with fractions is 'cross-multiplication.'

Here's how cross-multiplication works for an equation like \( \frac{1}{6x} = \frac{2}{x-6} \):

• Cross-multiplication eliminates the fractions by multiplying the numerator of each fraction by the denominator of the other.
• This means you set up the equation as: \( 1(x-6) = 2(6x) \).
• Simplifying this, you get \( x - 6 = 12x \).

By continued simplification, you subtract \( x \) from both sides, and then divide by the remaining coefficient to solve for \( x \). This method effectively isolates the variable, offering a straightforward solution process.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator contain polynomials. Understanding them is key to tackling intermediate algebra.

In these expressions, operations like addition, subtraction, multiplication, or division often require simplification. Let's see the basics needed when working with rational expressions:

• First, factor both numerators and denominators whenever possible to find common terms.
• Simplification may involve cancelling out these terms, reducing the rational expression's complexity.
• With expressions like \( \frac{-11x - 6}{6x(x-6)} \), always check if further simplification is possible by factoring the numerator.

Mastering rational expressions involves recognizing patterns and applying consistent simplification rules, making them more manageable and opening the path to easier equation solving.