Problem 72 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{\frac{x+h}{x+h+1}-\frac{x}{x+1}}{h}$$

Step-by-Step Solution

Answer

The simplified version of the given complex rational expression is \(\frac{1-x}{(x+1)(x+h+1)}\).

Step 1: Simplify the numerators

Find a common denominator for the two fractions in the numerator and simplify. The common denominator is \((x+h+1)(x+1)\), so the equation becomes: \[\frac{\frac{(x+h)(x+1)-(x)(x+h+1)}{(x+h+1)(x+1)}}{h}\]

Step 2: Simplify the Numerator

Combine like terms in the numerator. This results in: \[\frac{x^2+x+hx+h-x^2-x-hx-1-xh}{(x+h+1)(x+1)} \cdot \frac{1}{h}\] Simplifying this results in: \[\frac{h-hx}{(x+h+1)(x+1)} \cdot \frac{1}{h}\]

Step 3: Simplify the Complex Fraction

Now, simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator. When we cancel out the \(h\) terms from numerator with denominator, we are left with: \[\frac{1-x}{(x+1)(x+h+1)}\]