Add the following rational expressions.
$\frac{2 x+3}{x+2}+\frac{x-2}{x+2}=$
The task is to perform the addition of two rational expressions: \(\frac{2x+3}{x+2} + \frac{x-2}{x+2}\). First, we check if the expressions are well-defined. Since the denominator is \(x+2\), we must ensure that \(x+2 eq 0\), which implies \(x eq -2\). Given this condition, we can proceed with the addition.
To add two fractions with the same denominator, we use the rule:
\[\frac{A}{C} + \frac{B}{C} = \frac{A+B}{C}.\]
In this case, \(A = 2x+3\), \(B = x-2\), and \(C = x+2\). Applying the rule, we get:
\[\frac{2x+3}{x+2} + \frac{x-2}{x+2} = \frac{(2x+3) + (x-2)}{x+2}.\]
Now, simplify the numerator by combining like terms. The expression in the numerator is \((2x+3) + (x-2)\).
Thus, the numerator becomes \(3x + 1\). The entire expression is now \(\frac{3x+1}{x+2}\).
The simplified expression is \(\frac{3x+1}{x+2}\), with the condition \(x eq -2\). There are no further common factors between the numerator and denominator to simplify. Therefore, the final result is \(\boxed{\frac{3x+1}{x+2}}\).
Adding rational expressions is very useful in fields like engineering and physics when dealing with complex circuits or variable forces. For instance, if you are calculating the total effective resistance in a circuit where components have variable resistances depending on current or temperature, you might need to sum fractional expressions. Understanding how to simplify these helps in reducing the complexity of the equations representing the system, making them easier to solve for specific physical parameters.