Problem 92 - Elementary Algebra

Question

For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Simplify: \(\frac{c}{c-1}+\frac{4}{c+2}\) Incorrect Answer: The least common denominator is \((c-1)(c+2)\). $$ \begin{aligned} &\frac{c}{c-1}+\frac{4}{c+2} \\ &=\frac{c}{c-1} \cdot \frac{(c+2)}{(c+2)}+\frac{4}{c+2} \cdot \frac{(c-1)}{(c-1)} \\ &=\frac{c^{2}+2 c}{(c-1)(c+2)}+\frac{4 c-4}{(c-1)(c+2)} \\ &=\frac{c^{2}+6 c-4}{(c-1)(c+2)} \\ &=\frac{c^{2}+6 c-4}{c^{2}+c-2} \\ &=\frac{6 c-4}{c-2} \end{aligned} $$

Step-by-Step Solution

Answer

The correct simplified form is \[ \frac{c^2 + 6c - 4}{(c-1)(c+2)} \].

Step 1: Identify the Error

Review the problem and incorrect solution. Notice where the incorrect computations or operations occurred. In this problem, the mistake happens during the simplification process after combining the fractions over the common denominator.

Step 2: Determine the Correct Least Common Denominator (LCD)

Identify the least common denominator for the given fractions: \[ \frac{c}{c-1} + \frac{4}{c+2} \] The correct LCD is indeed \((c-1)(c+2)\).

Step 3: Rewrite Each Fraction with the LCD

Adjust each fraction so that they have the common denominator, expressing each fraction with this new denominator: \[\frac{c}{c-1} \times \frac{(c+2)}{(c+2)} = \frac{c(c+2)}{(c-1)(c+2)} = \frac{c^2 + 2c}{(c-1)(c+2)} \] and \[ \frac{4}{c+2} \times \frac{(c-1)}{(c-1)} = \frac{4(c-1)}{(c-1)(c+2)} = \frac{4c - 4}{(c-1)(c+2)} \]

Step 4: Add the Fractions

Combine the two adjusted fractions over the common denominator: \[\frac{c^2 + 2c}{(c-1)(c+2)} + \frac{4c - 4}{(c-1)(c+2)} = \frac{c^2 + 2c + 4c - 4}{(c-1)(c+2)} \]

Step 5: Simplify the Numerator

Combine like terms in the numerator: \[ c^2 + 2c + 4c - 4 = c^2 + 6c - 4 \] The combined fraction becomes: \[ \frac{c^2 + 6c - 4}{(c-1)(c+2)} \]

Step 6: Verify the Final Expression

Ensure the numerator and denominator do not simplify further: The correct simplified form is: \[ \frac{c^2 + 6c - 4}{(c-1)(c+2)} \]

Step 7: Conclusion

The mistake was made in the incorrect simplification beyond this point. The error was in wrongly reducing the fraction incorrectly in the provided solution. The problem is now correctly solved.

Key Concepts

least common denominator

When adding algebraic fractions like \(\frac{c}{c-1}\frac{4}{c+2}\), we need a common denominator. This common denominator, also known as the Least Common Denominator (LCD), allows us to write both fractions with the same base.

To find the LCD of \(\frac{c}{c-1}\frac{4}{c+2}\), we multiply the distinct denominators: \(c-1\) and \(c+2\). Hence, our LCD is \( (c-1)(c+2) \).

Using the LCD, we can rewrite each fraction:

• \text{\frac{c}{c-1} becomes \frac{c(c+2)}{(c-1)(c+2)} = \frac{c^2 + 2c}{(c-1)(c+2)}}\boxed


• \text{\frac{4}{c+2} becomes \frac{4(c-1)}{(c+2)(c-1)} = \frac{4c-4}{(c+1)(c+2)}}\boxed

Understanding LCD is crucial for correctly performing operations on algebraic fractions.

simplifying fractions

Once we have the fractions with the same denominator, the next step is to add or subtract the numerators. After combining the fractions, we obtain:

\begin{aligned}\frac{c^2 + 2c}{(c-1)(c+2)} + \frac{4c - 4}{(c-1)(c+2)} becomes \frac{c^2 + 2c + 4c - 4}{(c-1)(c+2)} \boxed\begin{aligned}\br> This step involves combining the numerators over the common denominator.

Next, we need to simplify the numeric fraction as much as possible. Combine the like terms and simplify if possible:
\begin{aligned}\frac{c^2 + 2c + 4c - 4}{(c-1)(c+2)} = \frac{c^2 + 6c - 4}{(c-1)(c+2)}\boxed\begin{aligned}\br>Always make sure to check if the numerator and denominator can be further reduced, but in this case, the fraction \(\frac{c^2 + 6c - 4}{(c-1)(c+2)}\) cannot be simplified further.
Simplifying fractions helps in obtaining the most reduced form for better understanding and clarity.

combining like terms

Combining like terms is essential in algebra when simplifying expressions. Here, after finding the LCD and rewriting the fractions, we combine the like terms in the numerator.\br\br>For example:\br\begin{aligned} c^2 + 2c + 4c - 4 = c^2 + 6c - 4\boxed\br> We group similar terms together:

• \br>Combine \(2c\) and \(4c\) to get \(6c\).\br\br>Remember to always look for and group similar terms - terms that have the same variable raised to the same power. \br\br- This helps in simplifying algebraic expressions to their most reduced form for more straightforward calculations and easier understanding.