Problem 98 - Introductory Algebra for College Students

Question

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}-13}{x+4}-\frac{3}{x+4}=x+4, x \neq-4$$

Step-by-Step Solution

Answer

The subtraction operation is incorrect. The correct operation should have been \((x^{2} - 16[x+4])/(x+4)\) which simplifies to \(x+4= x+4\), assuming \(x\neq-4\). This can be verified using a graphing utility which will show that these two expressions are equal for all \(x \neq -4\) values.

Step 1: Simplify The Equation

First, simplify the left-hand side of the equation by combining the two fractions. To do this, note that the two fractions already have a common denominator, \(x+4\). Hence, their subtraction can be carried out normally as follows: \[ \frac{x^{2} - 13 - 3}{x+4} = \frac{x^{2} - 16}{x+4} = x + 4 \]

Step 2: Compare Both Sides

Next, on comparing the simplified left-hand side with the right-hand side, it can be observed that they do not match. Thus, the original subtraction seems to be incorrect.

Step 3: Correct The Subtraction

The corrected subtraction operation would look like: \[ \frac{x^{2} - 16}{x+4} \neq x+4 \] because the result \(x+4\) on the right-hand side would be the correct answer if the numerator was \((x^2 - 16[x+4])\) instead of \((x^2 - 16)\)

Step 4: Verify Using a Graphing Utility

You can use a graphing utility to plot the equations: \[ y = \frac{x^{2} - 16}{x+4} \] and \[ y = x+4 \] If the subtraction had been performed correctly, both plots would overlap. However, for the proposed equation, the functions plotted will not coincide, indicating that the subtraction was incorrect.

Key Concepts

Algebra

Algebra serves as a fundamental building block in mathematics. It involves working with symbols and the rules for manipulating these symbols to solve equations and express general relationships between quantities. The essence of algebra is to solve equations and find unknown values.

• Variables: These are symbols that represent numbers whose values are not yet known. For example, x in an equation.
• Expressions: A combination of variables, constants, and operators (like +, −, ×, ÷).
• Equations: A mathematical statement that asserts the equality of two expressions. It is solved to find the values of the variables that make the equation true.

Understanding the rules and operations in algebra allows us to simplify equations effectively, as seen when combining like terms or simplifying expressions with common denominators.

Polynomial Division

Polynomial division, akin to long division with numbers, is the process of dividing one polynomial by another polynomial. The quotient obtained may be another polynomial, plus a remainder.

• Divisor: The polynomial we divide by.
• Dividend: The polynomial we wish to divide.
• Quotient: The resulting polynomial from the division.
• Remainder: The leftover part of the dividend that's not divisible by the divisor.

In the original exercise, polynomial division is applied implicitly when simplifying the subtraction of two rational expressions. Each part's numerator is adjusted while the common denominator is maintained, turning complex expressions into simpler forms with algebraic clarity.

Graphing Utility

A graphing utility is a digital tool used to plot functions and visualize mathematical concepts. It is invaluable for checking equations, observing function behavior, and analyzing where two graphs may intersect.

• Plotting Functions: Allows you to view the visual representation of equations.
• Intersect Points: Helps identify solutions to equations where graphs intersect.
• Trend Analysis: Easily observe how changes in equations affect the graph.

Using a graphing utility in the exercise, students can verify their mathematical operations by visually confirming if different sides of an equation overlap, showing equality, or if they diverge, indicating an error.

Equation Solving

Solving equations is the process of finding the value(s) of variable(s) that satisfy the equation. The goal is to isolate the variable on one side of the equation.

• Simplification: Often necessary to combine like terms and make the equation easier to solve.
• Substitution: Replacing a variable with a number or another expression to test potential solutions.
• Checking Solutions: Once solved, it is important to verify by substituting back into the original equation.

In the exact exercise, the key task involves simplifying both sides and recognizing errors. When the simplified left side didn't match the right side, the student was prompted to identify mistakes in the solution. Graphing utilities then helped check this algebra work by plotting the original and simplified equations to see the divergence.