Solve the inequality and write the solution in interval notation.
$-24<\frac{4}{3} n$
The given inequality is \(-24 < \frac{4}{3}n\), where \(n\) is a real number. Our goal is to isolate the variable \(n\) to determine the range of values that satisfy this inequality. A linear inequality like this can be solved using algebraic operations, similar to solving a linear equation, with the caveat that multiplying or dividing both sides by a negative number reverses the inequality sign.
To isolate \(n\), we need to eliminate the coefficient \(\frac{4}{3}\) from the right side of the inequality. We do this by multiplying both sides of the inequality by the reciprocal of \(\frac{4}{3}\), which is \(\frac{3}{4}\). Since \(\frac{3}{4}\) is a positive number, the direction of the inequality sign remains unchanged.
The operation is:
\[-24 \times \frac{3}{4} < \left(\frac{4}{3}n\right) \times \frac{3}{4}\]
Simplifying the left side:
\[-24 \times \frac{3}{4} = -6 \times 3 = -18\]
Simplifying the right side:
\[\left(\frac{4}{3} \times \frac{3}{4}\right)n = 1 \times n = n\]
Thus, the inequality simplifies to:
\[-18 < n\]
or, expressed in terms of \(n\):
\[n > -18\]
The solution \(n > -18\) indicates that \(n\) can be any real number greater than \(-18\). In interval notation, we represent this set of all real numbers strictly greater than \(-18\) as \((-18, \infty)\). The parenthesis on the left indicates that \(-18\) is not included in the solution set, and the infinity symbol on the right indicates that the values continue indefinitely in the positive direction. The final solution is \(\boxed{(-18, \infty)}\).
In real-life scenarios, solving linear inequalities is useful for determining thresholds. For example, if a store needs to have a profit of more than \(\$1000\) to stay open, and their profit is modeled by a function \(P(n) = 20n - 500\), where \(n\) is the number of items sold, they would solve \(20n - 500 > 1000\) to find the minimum number of items they need to sell to reach that goal. This mathematical approach helps business owners make data-driven decisions about their operations and ensure they meet their targets.