Problem 93 - Introductory Algebra for College Students

Question

Use the GRAPH or TABLE feature of a graphing utility to determine if the multiplication or division has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4}=x-3$$

Step-by-Step Solution

Answer

The original division equation \( \frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4} = x-3 \) was incorrect. It should have been \( \frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4} = x-3 \). The graphing utility confirmed this correction.

Step 1: Set up the equation

First, check the given division operation: \( \frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4} = x-3 \) Since division is the same as multiplication by the reciprocal, the equation can be rewritten as: \( \frac{x^{2}-9}{x+4} \times \frac{x+4}{x-3} = x-3 \)

Step 2: Simplify the equation

Simplify the equation by multiplying the rational expressions, e.g. by cancelling equivalent terms in the numerator and denominator of the multiplication: \( \frac{(x-3)(x+3)}{x+4} \times \frac{x+4}{x-3} = x-3 \) This simplifies to: \( x+3 = x-3 \)

Step 3: Graph the equation

Graph the simplified equation \( x+3 = x-3 \) using a graphing utility. The graphing utility will clearly show where the solution is not consistent with the equation.

Step 4: Interpret the graph

Analyze the graph and it is seen that the lines do not intersect, thus there are no solutions. This indicates an error in the original equation.

Step 5: Correct the original equation

If the original equation was incorrect, correct it. Reconsider the operation and realize that when the first function is multiplied by the reciprocal of the second, the \( x+4 \) terms should cancel out, leaving only \( x-3 \) on both sides of the equation. Thus, the corrected division equation is: \( \frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4} = x-3 \)

Step 6: Verify the correction

Verify this correction by graphing the new equation and observing that the solution is consistent with the equation: \( x-3 = x-3 \)

Key Concepts

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division are essential to manipulate algebraic expressions. When dealing with rational expressions, division requires a special approach. Instead of dividing directly, we multiply by the reciprocal. This means

• For expression \[ \frac{a}{b} \div \frac{c}{d} \], you change it to \[ \frac{a}{b} \times \frac{d}{c} \].
• This step converts the division into a multiplication problem.
• By proceeding with multiplication, it becomes easier to simplify any mathematical expression.

Don't forget that cancelling common terms in the numerator and denominator simplifies these operations further.

Graphing Utilities

A graphing utility is an efficient tool that can help visualize complex mathematical equations, specifically in solving rational expressions. These tools

• Generate visual graphs where you can easily spot intersections, roots, or inconsistencies.
• Assist in verifying whether an algebraic solution is correct or needs adjustment.

For instance, when comparing two sides of an equation graphically, like \( x + 3 \) and \( x - 3 \), you can see they do not intersect at any point, confirming an error in calculation. This insight helps in either confirming your answer or pointing out precisely where corrections are needed.

Algebraic Simplification

Algebraic simplification is about making expressions easier to read and work with. This often involves the following steps:

• Factorizing expressions to identify and cancel out common terms.
• Simplifying terms within the expressions.
• Rewriting equations to their simplest form to allow easier analysis or further operations.

In our example, converting the expression from \( \frac{(x-3)(x+3)}{x+4} \times \frac{x+4}{x-3} \) to \( x + 3 = x - 3 \) shows how simplified forms can point out inconsistencies (i.e., featuring how an expression equates 0 or any false result). This process not only simplifies the operations but also highlights mistakes.

Multiplication and Division of Polynomials

Polynomials under multiplication and division require attention to detail, especially when dealing with rational expressions. This involves:

• Multiplying terms directly across numerators and denominators.
• Ensuring to factor polynomials to identify which terms can be simplified through cancellation.
• Checking for restrictions or points where the expression is undefined, such as where denominators would equal zero.

In the given exercise, the expression\( \frac{x^2-9}{x+4} \div \frac{x-3}{x+4} \) becomes\( \frac{x^2-9}{x+4} \times \frac{x+4}{x-3} \), illustrating the multiplication of polynomials, but also shows that not accounting for cancellations correctly can lead to mathematical errors. Understanding how to handle these expressions is crucial for correct solutions.