Questions: What is the difference of the rational expressions below? 9/x^2 - (2x+1)/(8x) A. (-2x+8)/(8x^2) B. (-2x^2-x+72)/(8x^2) C. (-2x^2+x+72)/(8x^2) D. (-2x+10)/(8x^2)

Transcript text: What is the difference of the rational expressions below? \[ \frac{9}{x^{2}}-\frac{2 x+1}{8 x} \] A. $\frac{-2 x+8}{8 x^{2}}$ B. $\frac{-2 x^{2}-x+72}{8 x^{2}}$ C. $\frac{-2 x^{2}+x+72}{8 x^{2}}$ D. $\frac{-2 x+10}{8 x^{2}}$

Solution

Solution Steps

Step 1: Find a Common Denominator

The given expression is: \[ \frac{9}{x^{2}} - \frac{2x + 1}{8x} \] The denominators are $ x^{2} $ and $ 8x $. The least common denominator (LCD) is $ 8x^{2} $.

Step 2: Rewrite Each Fraction with the LCD

Rewrite each fraction to have the denominator $ 8x^{2} $: \[ \frac{9}{x^{2}} = \frac{9 \cdot 8}{x^{2} \cdot 8} = \frac{72}{8x^{2}} \] \[ \frac{2x + 1}{8x} = \frac{(2x + 1) \cdot x}{8x \cdot x} = \frac{2x^{2} + x}{8x^{2}} \]

Step 3: Subtract the Fractions

Subtract the two fractions: \[ \frac{72}{8x^{2}} - \frac{2x^{2} + x}{8x^{2}} = \frac{72 - (2x^{2} + x)}{8x^{2}} = \frac{-2x^{2} - x + 72}{8x^{2}} \]