- To solve the inequality \(-5t \leq -30\), isolate \(t\) by dividing both sides by \(-5\).
- Because \(-5\) is a negative number, the inequality sign must be reversed according to the properties of inequalities.
- Applying this, we get \(t \geq 6\).
- The correct procedure is to divide both sides by \(-5\) and reverse the inequality symbol. \(\boxed{\text{Divide both sides of the inequality by -5 and reverse the inequality symbol.}}\)
Explanation
- Isolating the Variable
To solve the inequality \(-5t \leq -30\) for the variable \(t\), our primary objective is to isolate \(t\). We are given that \(-5\) is multiplied by \(t\). To isolate \(t\), we must perform the inverse operation, which is division by \(-5\).
- The Effect of Negative Division
A fundamental rule in algebra states that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This ensures that the mathematical relationship remains true. In our case, dividing \(-5t\) by \(-5\) gives \(t\), and dividing \(-30\) by \(-5\) gives \(6\). Because we are dividing by a negative number, the \(\leq\) sign must be flipped to \(\geq\).
- Conclusion
Following the rule identified in the previous step, we divide both sides by \(-5\) and change the direction of the inequality:
\[\frac{-5t}{-5} \geq \frac{-30}{-5}\]
\[t \geq 6\]
Thus, the correct procedure is to divide both sides by \(-5\) and reverse the inequality symbol.
Examples
Imagine you are tracking your spending against a budget. If you have a negative balance trend due to daily expenses, such as \(-5 \times (\text{days}) \leq -\$30\), this math helps you determine the number of days you must exceed or meet to stay within a limit. Just like in our problem, understanding when to flip the inequality sign is crucial for correctly interpreting resource constraints and planning your finances effectively.