Problem 38 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{1 - \frac{2}{x}}\), when \(x ≠ 0\).

Step 1: 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.

Step 2: Subtract the numerators

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

Step 3: 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: \(\frac{{3x}}{{3x - 6}}\).

Step 4: 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 \(x ≠ 0\), this simplifies to: \(\frac{1}{1 - \frac{2}{x}}\).