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Chapter 6: Problem 60

Add or subtract as indicated. $$\frac{3}{x-3}-\frac{1}{x}$$

Short Answer

Expert verified
\frac{2x+3}{x(x-3)}

Step by step solution

01

Determine the common denominator

To subtract the fractions, a common denominator is needed. For \(\frac{3}{x-3} - \frac{1}{x}\), the common denominator is \(x(x-3)\).
02

Rewrite each fraction with the common denominator

Convert each fraction so they both have the common denominator \(x(x-3)\): \(\frac{3}{x-3} = \frac{3x}{x(x-3)}\) and \(\frac{1}{x} = \frac{x-3}{x(x-3)}\).
03

Subtract the fractions

Now subtract the fractions: \(\frac{3x}{x(x-3)} - \frac{x-3}{x(x-3)} = \frac{3x-(x-3)}{x(x-3)}\).
04

Simplify the numerator

Simplify the expression in the numerator: \(\frac{3x - (x - 3)}{x(x-3)} = \frac{3x - x + 3}{x(x-3)} = \frac{2x + 3}{x(x-3)}\).

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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 is crucial to have the same denominator for both fractions. This common denominator acts as a base that allows us to combine the fractions. Think of it like having the same measuring unit.
For algebraic fractions, identify the least common multiple (LCM) of the denominators. Here, we have \(\frac{3}{x-3} - \frac{1}{x}\). The denominators are \((x - 3)\) and \(x\).
To find the common denominator, we multiply the denominators of each fraction together: \(x(x - 3)\).
Simplifying Fractions
Simplifying fractions means making them as simple as possible.
We start by converting each fraction to have the common denominator obtained earlier. For \(\frac{3}{x-3}\), we multiply both the numerator and denominator by \(x\), giving us \(\frac{3x}{x(x-3)}\).
For \(\frac{1}{x}\), we multiply the numerator and denominator by \(x-3\), giving \(\frac{x-3}{x(x-3)}\).
Hence, both fractions have the same denominator: \(x(x-3)\).
Subtracting Fractions
Once the fractions have a common denominator, we can subtract them.
Using our fraction example \(\frac{3}{x-3} - \frac{1}{x}\), now written as \(\frac{3x}{x(x-3)} - \frac{x-3}{x(x-3)}\), we can subtract the numerators: \( \frac{3x-(x-3)}{x(x-3)} = \frac{3x-x+3}{x(x-3)} = \frac{2x+3}{x(x-3)}\).
Always combine the numerators, leaving the common denominator unchanged.
Algebraic Expressions
In fractions involving algebraic expressions, it鈥檚 important to handle variables appropriately.
Fractions like \(\frac{3}{x-3}\) and \(\frac{1}{x}\) involve variables \(x\) that change the fraction's value based on their value.
When performing operations on such expressions, always simplify by treating the variables carefully and expanding or factoring algebraic expressions to keep terms clear and manageable.

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Most popular questions from this chapter

Which two rational expressions are equivalent to \(-1 ?\) A. \(\frac{2 x+3}{2 x-3}\) B. \(\frac{2 x-3}{3-2 x}\) C. \(\frac{2 x+3}{3+2 x}\) D. \(\frac{2 x+3}{-2 x-3}\)

Write each rational expression in lowest terms. $$ \frac{(x+4)(x-3)}{(x+5)(x+4)} $$

Multiply or divide as indicated. $$ \left(\frac{6 k^{2}-13 k-5}{k^{2}+7 k} \div \frac{2 k-5}{k^{3}+6 k^{2}-7 k}\right) \cdot \frac{k^{2}-5 k+6}{3 k^{2}-8 k-3} $$

Write each rational expression in lowest terms. $$ \frac{4 x(x+3)}{8 x^{2}(x-3)} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{3}{8} \div \frac{9}{14} $$
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