$\frac{x+3}{x-3}+\frac{x+8}{x-7}$
Simplify your answer as much as possible.
The objective is to simplify the sum of two rational expressions: \(\frac{x+3}{x-3} + \frac{x+8}{x-7}\). The expression is defined for all real numbers \(x\) except \(x = 3\) and \(x = 7\), as these values would result in division by zero. To simplify the sum, we must combine the two fractions into a single rational expression by finding a common denominator.
To add the two fractions \(\frac{x+3}{x-3}\) and \(\frac{x+8}{x-7}\), we identify the least common denominator as the product of the denominators: \((x-3)(x-7)\). We rewrite each fraction with this denominator:
Now, add the two fractions by combining their numerators over the common denominator:
\[\frac{x^2 - 4x - 21 + x^2 + 5x - 24}{(x-3)(x-7)}\]
Combining like terms in the numerator:
The numerator is \(2x^2 + x - 45\). Thus, the expression is:
\[\frac{2x^2 + x - 45}{(x-3)(x-7)}\]
To check if the expression can be simplified further, we attempt to factor the numerator \(2x^2 + x - 45\). We look for two numbers that multiply to \(2 \times (-45) = -90\) and add to \(1\). These numbers are \(10\) and \(-9\).
\[2x^2 + 10x - 9x - 45 = 2x(x + 5) - 9(x + 5) = (2x - 9)(x + 5)\]
The simplified expression is \(\frac{(2x - 9)(x + 5)}{(x-3)(x-7)}\). Since no factors cancel with the denominator \((x-3)(x-7)\), this is the fully simplified form.
Simplifying rational expressions is fundamental when analyzing physical systems or economic models. For instance, if you are modeling the average cost per unit in a production process, you might encounter expressions where two different cost components (like fixed costs and variable costs) are represented by rational functions of the production volume \(x\). Combining these into a single expression allows you to quickly find the total average cost, analyze its behavior as production scales, and identify optimal operating points where cost is minimized.