Problem 11 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{x}{x-4}-\frac{2 x}{x+4}=-1 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{4}{3} \).

Step 1: Identify the Common Denominator

The given equation is \( \frac{x}{x-4} - \frac{2x}{x+4} = -1 \). To solve this, we first need to find a common denominator to combine the fractions. The denominators are \( x-4 \) and \( x+4 \). Therefore, the common denominator is \( (x-4)(x+4) \).

Step 2: Rewrite Each Fraction

Rewrite each fraction with the common denominator: \( \frac{x(x+4)}{(x-4)(x+4)} - \frac{2x(x-4)}{(x-4)(x+4)} \).

Step 3: Combine the Fractions

Since both fractions now have the same denominator, we can combine them: \( \frac{x(x+4) - 2x(x-4)}{(x-4)(x+4)} = -1 \).

Step 4: Simplify the Numerator

Distribute and simplify the numerator: \( x^2 + 4x - (2x^2 - 8x) = x^2 + 4x - 2x^2 + 8x = -x^2 + 12x \). So the equation becomes \( \frac{-x^2 + 12x}{(x-4)(x+4)} = -1 \).

Step 5: Eliminate the Denominator

To eliminate the denominator, multiply both sides of the equation by \( (x-4)(x+4) \) to get \( -x^2 + 12x = - (x-4)(x+4) \).

Step 6: Apply the Difference of Squares

Use the difference of squares formula: \( (x-4)(x+4) = x^2 - 16 \). Substitute in to get \( -x^2 + 12x = -x^2 + 16 \).

Step 7: Solve for x

Set the equation to zero and solve: \( -x^2 + 12x + x^2 - 16 = 0 \), simplifying to \( 12x - 16 = 0 \). Solve for \( x \) by adding 16 to both sides, yielding \( 12x = 16 \), then divide by 12 to find \( x = \frac{4}{3} \).

Step 8: Verify the Solution

Verify that \( x = \frac{4}{3} \) is not an extraneous solution by checking it doesn't make any denominator zero: \( x-4 eq 0 \) and \( x+4 eq 0 \) for \( x = \frac{4}{3} \).

Key Concepts

Common Denominator

When solving equations that involve fractions, finding a common denominator is essential. In simple terms, a common denominator allows different fractions to be expressed over the same base, making addition or subtraction possible.
To illustrate, consider this equation: \( \frac{x}{x-4} - \frac{2x}{x+4} = -1 \). Here, the denominators are \( x-4 \) and \( x+4 \).
Identifying the common denominator involves:

• Taking both denominators \( (x-4) \) and \( (x+4) \)
• Multiplying them together to get \( (x-4)(x+4) \)

Why is this important? Having a common denominator simplifies the equation because it allows you to directly compare and combine fractions. Without this step, it is much harder to simplify the expression or arrive at a solution.
Once you've rewritten each term to have the common denominator, simplification can begin. This makes solving such algebraic equations simpler and more straightforward.

Difference of Squares

The difference of squares is a fundamental algebraic identity that is often used to simplify expressions. It is expressed as \( a^2 - b^2 = (a-b)(a+b) \).
In our specific problem, we see the expression \( (x-4)(x+4) \) appear, which is a textbook example of the difference of squares:

• Here, \( a = x \) and \( b = 4 \).
• Using the identity, \( x^2 - 16 \) simplifies to \( (x-4)(x+4) \).

The use of this formula is crucial when dealing with such polynomial expressions. It not only simplifies the calculations but also turns a seemingly complex polynomial into a more easily manageable product of binomials.
Understanding this helps students to both recognize patterns and solve more intricate algebraic equations quickly.

Simplifying Expressions

Simplifying an algebraic expression is a skill that involves condensing it to its simplest form. This process makes the evaluation of equations more straightforward, as fewer terms are involved.
For example, in our initial problem, after finding a common denominator and applying the difference of squares, the expression becomes \( \frac{-x^2 + 12x}{(x-4)(x+4)} = -1 \). To simplify further, tackle the numerator first:

• Distribute and combine like terms: \( x^2 + 4x - 2x^2 + 8x = -x^2 + 12x \).

Here, negative signs and distribution play a key role in simplifying \( x(x+4) - 2x(x-4) \) into \( -x^2 + 12x \).
Remember, simplification involves:

• Combining like terms
• Reducing the number of operations
• Making the expression easy to solve or comprehend

By reducing complexity, similar steps can be applied to any other algebraic equation, consistently leading to clearer and more concise expressions.