Solve the inequality and write the solution in interval notation.
$-7r \leq -70$
We are asked to solve the linear inequality \(-7r \leq -70\) for the variable \(r\). Our primary goal is to isolate \(r\) on one side of the inequality. We must be careful because the coefficient of \(r\) is a negative number. One of the fundamental rules of working with inequalities is that if we multiply or divide both sides by a negative number, the direction of the inequality sign must be reversed to maintain the truth of the statement.
To isolate \(r\), we divide both sides of the inequality \(-7r \leq -70\) by \(-7\). As we are dividing by a negative number, we reverse the inequality symbol from \(\leq\) to \(\geq\). This gives us:
\[r \geq \frac{-70}{-7}\]
Performing the division on the right side:
\[\frac{-70}{-7} = 10\]
Thus, the simplified inequality is \(r \geq 10\).
The solution \(r \geq 10\) indicates that all real numbers greater than or equal to 10 are solutions to the original inequality. In interval notation, this is written as \([10, \infty)\), where the square bracket indicates that 10 is included in the solution set, and the parenthesis indicates that the interval extends to infinity.
Suppose you have a budget of \$70 that you need to spend on supplies for a project. Each unit of supply costs \$7, and you have to pay a flat fee that results in a debt reduction. If the expression -7r represents your current balance relative to a specific baseline, you must ensure that your total debt doesn't exceed -70 (or that your balance is at least -70). Solving this inequality helps you determine the minimum quantity \(r\) of supplies you must manage to stay within your required financial limits. This is a foundational skill for resource management and budgeting in everyday life.