What is the solution set for $-18 \leq-\frac{2}{3} t$ ?
We are given the linear inequality \(-18 \leq -\frac{2}{3} t\). Our goal is to isolate the variable \(t\) to determine its possible range. In algebra, inequalities behave similarly to equations, but with a critical difference: when multiplying or dividing both sides by a negative number, the direction of the inequality sign must be reversed. This is a fundamental rule that we must apply in our solution.
To solve for \(t\), we first look to isolate \(t\) on one side. We can achieve this by multiplying both sides of the inequality by the reciprocal of the coefficient of \(t\), which is \(-\frac{3}{2}\). Starting with:
\[-18 \leq -\frac{2}{3} t\]
Multiplying both sides by \(-\frac{3}{2}\):
\[-18 \times \left(-\frac{3}{2}\right) \geq -\frac{2}{3} t \times \left(-\frac{3}{2}\right)\]
Note that the inequality sign flipped from \(\leq\) to \(\geq\) because we multiplied by a negative value. Simplifying both sides, we get:
\[\frac{18 \times 3}{2} \geq t\]
\[9 \times 3 \geq t\]
\[27 \geq t\]
The inequality \(27 \geq t\) is equivalent to writing \(t \leq 27\). Looking at the options provided:
A \(t \geq 27\)
B \(t \leq 27\)
C \(t \geq 12\)
D \(t \leq 12\)
E none of these
Our result \(t \leq 27\) matches option B.
Understanding how to manipulate inequalities is vital for range-based calculations, such as setting temperature safety limits for a chemical process. If a thermometer sensor outputs an electrical signal that follows the relationship \(V \leq 5\) Volts for safety, and the signal is processed by a circuit such as \(-10 \leq -2V\), solving this inequality correctly ensures you stay within the specified safe operating parameters. Mastering these steps helps in ensuring accurate safety protocols in engineering and real-world monitoring systems.