Subtract.
$\frac{b+4}{2b} - \frac{b-5}{7b}$
Simplify your answer as much as possible.
We are tasked with simplifying the expression \(\frac{b+4}{2b} - \frac{b-5}{7b}\). To subtract fractions with different denominators, we must first find a common denominator. The denominators here are \(2b\) and \(7b\). The least common denominator (LCD) for \(2b\) and \(7b\) is \(14b\), since \(14b\) is the smallest expression divisible by both \(2b\) and \(7b\). We assume \(b eq 0\) for the expression to be well-defined.
To express both fractions with the common denominator of \(14b\), we multiply the numerator and denominator of each fraction by the necessary factor:
\[\frac{b+4}{2b} \cdot \frac{7}{7} = \frac{7(b+4)}{14b} = \frac{7b + 28}{14b}\]
\[\frac{b-5}{7b} \cdot \frac{2}{2} = \frac{2(b-5)}{14b} = \frac{2b - 10}{14b}\]
Now that both fractions have the same denominator, we can subtract them:
\[\frac{7b + 28}{14b} - \frac{2b - 10}{14b} = \frac{(7b + 28) - (2b - 10)}{14b}\]
Distribute the negative sign across the second part of the numerator:
\[\frac{7b + 28 - 2b + 10}{14b}\]
Combine like terms:
\[\frac{(7b - 2b) + (28 + 10)}{14b} = \frac{5b + 38}{14b}\]
Thus, the simplified form of the expression is \(\frac{5b + 38}{14b}\).
In real-life scenarios, simplifying rational expressions is a fundamental skill in engineering and physics when dealing with formulas involving variable quantities, such as calculating the net rate of flow in a pipe system or balancing chemical concentrations. For example, if you have two pumps moving water through a system at rates \(\frac{b+4}{2b}\) and \(\frac{b-5}{7b}\) liters per second, subtracting these expressions allows you to find the net change in water volume over time, ensuring the system operates correctly under different flow conditions.