Problem 72 - College Algebra

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 expression is \( \frac{1}{(x+h+1)(x+1)}\).

Step 1: Find Common Denominator

The first step is to find a common denominator for the fractional expressions in the numerator. This is done by multiplying the denominators of the two fractions together, which results in \((x+h+1)(x+1)\). Now, rewrite the fractional expressions in the numerator with the common denominator: \(\frac{(x+h)(x+1)-(x)(x+h+1)}{(x+h+1)(x+1)}\). Hence, the whole fraction becomes \(\frac{\frac{(x+h)(x+1)-(x)(x+h+1)}{(x+h+1)(x+1)}}{h}\).

Step 2: Simplifying the numerator

Next step is to distribute and combine like terms in the numerator. So the numerator becomes \(\frac{x^2+xh+x+h-x^2-xh-x}{(x+h+1)(x+1)}\) which simplifies to \( \frac{h}{(x+h+1)(x+1)}\). Therefore, the whole fraction simplifies to \( \frac{\frac{h}{(x+h+1)(x+1)}}{h}\).

Step 3: Cancel out the \(h\)

The final step is to cancel out \(h\) from the numerator and denominator so that \(h/h = 1\). So the final result of simplification is \( \frac{1}{(x+h+1)(x+1)}\).