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Chapter 7: Problem 23

Add or subtract. Write answer in lowest terms. \(\frac{x}{x^{2}-9}-\frac{-3}{x^{2}-9}\)

Short Answer

Expert verified
\( \frac{1}{x-3} \)

Step by step solution

01

Identify the Common Denominator

Both fractions have the same denominator: \(x^2 - 9\). Note that \(x^2 - 9 = (x + 3)(x - 3)\).
02

Combine the Numerators

Combine the numerators over the common denominator. The expression becomes: \( \frac{x}{(x + 3)(x - 3)} - \frac{-3}{(x + 3)(x - 3)} \). This simplifies to \( \frac{x + 3}{(x + 3)(x - 3)} \).
03

Simplify the Expression

Since the numerator \(x + 3\) is a factor of the denominator, they cancel out, leaving: \( \frac{1}{(x - 3)} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When adding or subtracting fractions, it鈥檚 crucial to work with a common denominator. This simplifies the process.
The denominator is the bottom part of a fraction. It tells us into how many parts the whole is divided. In our problem, both fractions have the same denominator: \(x^2 - 9\).
To factorize and understand better, we write it as: \(x^2 - 9 = (x + 3)(x - 3)\).
Both fractions now have \( (x + 3)(x - 3)\) as their common denominator.
Working with a common denominator allows us to easily combine or subtract the fractions.
Numerator and Denominator
In a fraction, the numerator is the top part, and the denominator is the bottom part. The numerator indicates how many parts we have, while the denominator indicates into how many parts the whole is divided.
In the problem \(\frac{x}{(x + 3)(x - 3)} -\frac{-3}{(x + 3)(x - 3)}\), we have:
  • \(\frac{x}{(x + 3)(x - 3)}\)
  • \(\frac{-3}{(x + 3)(x - 3)}\)

Since the denominators are identical, we can combine the numerators directly:
Numerators: \(x - (-3) = x + 3\)
The new fraction becomes: \(\frac{x + 3}{(x + 3)(x - 3)} \)
This step simplifies operations, making calculations and further simplifications more straightforward.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. We do this by canceling out common factors in the numerator and the denominator.
In the simplified fraction \(\frac{x + 3}{(x + 3)(x - 3)} \), the numerator is \(x + 3\), a factor also present in the denominator.
We can cancel out \(x + 3\) from both the numerator and the denominator. This leaves us with:
\(\frac{1}{x - 3}\)
Simplifying fractions helps in comprehending and comparing fractions more effortlessly. It also aids in further mathematical operations.

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Most popular questions from this chapter

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{3}{7} \cdot \frac{4}{5} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{4 m}{m^{2}+m-2}=\frac{?}{(m-1)(m-3)(m+2)} $$

Solve each variation problem. Over a specified distance, rate varies inversely with time. If a Nissan \(370 \mathrm{Z}\) on a test track goes a certain distance in one-half minute at \(160 \mathrm{mph},\) what rate is needed to go the same distance in three- fourths minute? (IMAGE CAN'T COPY)

Solve each formula for \(k\) $$ y=\frac{k}{x^{2}} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{4 r-t}{r^{2}+r t+t^{2}}=\frac{?}{t^{3}-r^{3}} $$
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