Problem 41 - Algebra A Combined Function

Question

Solve the following. Two divided by the difference of a number and 3 minus 4 divided by the number plus \(3,\) equals 8 times the reciprocal of the difference of the number squared and \(9 .\) What is the number?

Step-by-Step Solution

Answer

The number is 5.

Step 1: Define the Equation

First, translate the word problem into a mathematical expression based on the description given. Let the unknown number be represented by \( x \). The problem states:\[ \frac{2}{x - 3} - \frac{4}{x + 3} = 8 \times \frac{1}{x^2 - 9} \]

Step 2: Recognize a Difference of Squares

Notice that \( x^2 - 9 \) can be factored as a difference of squares:\[ x^2 - 9 = (x - 3)(x + 3) \]This understanding will assist in simplifying the equation.

Step 3: Substitute and Simplify

Substitute \( \frac{1}{x^2 - 9} \) with \( \frac{1}{(x - 3)(x + 3)} \):\[ \frac{2}{x - 3} - \frac{4}{x + 3} = \frac{8}{(x - 3)(x + 3)} \]

Step 4: Find a Common Denominator

To combine the terms on the left side of the equation, determine a common denominator. The common denominator for the fractions \( \frac{2}{x-3} \) and \( \frac{4}{x+3} \) as well as on the right side is \((x - 3)(x + 3)\). Rewrite each term:\[ \frac{2(x+3)}{(x - 3)(x + 3)} - \frac{4(x-3)}{(x - 3)(x + 3)} = \frac{8}{(x - 3)(x + 3)} \]

Step 5: Simplify the Numerators

Simplify each numerator:\[ 2(x + 3) = 2x + 6 \]\[ -4(x - 3) = -4x + 12 \]This leads to simplified numerators:\[ \frac{2x + 6 - 4x + 12}{(x - 3)(x + 3)} = \frac{8}{(x - 3)(x + 3)} \]

Step 6: Combine Like Terms

Combine like terms in the numerator:\[ (2x - 4x) + (6 + 12) = -2x + 18 \]The equation now is:\[ \frac{-2x + 18}{(x - 3)(x + 3)} = \frac{8}{(x - 3)(x + 3)} \]

Step 7: Cross-Multiply and Solve for x

Cross-multiply to eliminate the fractions:\[ -2x + 18 = 8 \]Rearrange the equation to solve for \( x \): \[ -2x = 8 - 18 \]\[ -2x = -10 \]Divide both sides by \(-2\):\[ x = 5 \]

Step 8: Verify the Solution

Substitute \( x = 5 \) back into the original statement to verify it satisfies the equation. Check using simplification:\[ \frac{2}{5 - 3} - \frac{4}{5 + 3} = 8 \times \frac{1}{5^2 - 9} \]Simplify:\[ 1 - \frac{1}{2} = \frac{1}{2} \] The solution satisfies the given equation.

Key Concepts

Difference of Squares

Understanding the difference of squares concept is crucial when dealing with quadratic expressions. The difference of squares is a mathematical identity expressed as: \( a^2 - b^2 = (a - b)(a + b) \). This identity shows that a squared number minus another squared number can be factored into the product of two binomials. In simpler terms, when you see an equation like \( x^2 - 9 \), it's signaling that you can break it down into more manageable parts:

• Identify \( a = x \) and \( b = 3 \)
• Therefore, \( x^2 - 9 = (x - 3)(x + 3) \)

This factoring helps simplify complex rational equations by making it easier to find common denominators or simplify expressions. It is incredibly helpful in algebraic problem-solving and transforms a challenging quadratic into two simpler linear factors.

Rational Equations

Rational equations are equations containing at least one fraction whose numerator and/or denominator is a polynomial. Solving rational equations often involves finding a common denominator to eliminate the fractions and then solving the resulting polynomial equation. Let's take a closer look at this through our original equation: \[ \frac{2}{x - 3} - \frac{4}{x + 3} = \frac{8}{x^2 - 9} \] Here, the denominators \( x - 3 \), \( x + 3 \), and \( x^2 - 9 \) need to be considered. First, you must recognize \( x^2 - 9 \) as a difference of squares; thus, \((x - 3)(x + 3)\) becomes the common denominator. By rewriting each term with this common denominator, the fractions are effectively combined. This helps simplify the entire rational equation into a linear equation, which is easier to solve.

Cross-Multiplication

Cross-multiplication is a powerful algebraic technique used to eliminate fractions in an equation, particularly useful when working with rational equations. In this method, one multiplies across the equal sign diagonally—numerator of one side with the denominator of the other, and vice-versa. In our detailed solution, after simplifying the equation to: \[ \frac{-2x + 18}{(x - 3)(x + 3)} = \frac{8}{(x - 3)(x + 3)} \] We efficiently apply cross-multiplication. Since the denominators are identical, we can equate the numerators directly: \( -2x + 18 = 8 \) This step removes the fractions, leaving us with a straightforward linear equation to solve. By moving terms around and simplifying, the value of \( x \) can be found. Cross-multiplying is especially insightful when working through problems involving rational equations, making it a fundamental tool in algebra.