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Chapter 0: Problem 38

Add or subtract as indicated. $$\frac{2 x+3}{3 x-6}-\frac{3-x}{3 x-6}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{1 - \frac{2}{x}}\), when \(x 鈮� 0\).

Step by step solution

01

Identify the common denominator

Since the denominators of both fractions are the same, that is, \(3x - 6\), this is the common denominator. So no adjustments are needed before the operation.
02

Subtract the numerators

Subtract the second numerator from the first numerator. Thus the operation becomes: \(\frac{{2x + 3 - (3 - x)}}{{3x - 6}}\).
03

Simplify the numerator

By performing the subtraction in the numerator, we have: \(\frac{{2x + 3 - 3 + x}}{{3x - 6}}\), and combining like terms in the numerator gives: \(\frac{{3x}}{{3x - 6}}\).
04

Simplify the entire fraction

Observe that \(3x\) in the numerator and \(3x - 6\) in the denominator have a common factor of \(3x\). Dividing both the numerator and denominator by \(3x\) gives: \(\frac{1}{1 - \frac{6}{3x}}\). When \(x 鈮� 0\), this simplifies to: \(\frac{1}{1 - \frac{2}{x}}\).

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