Solve the following inequality for $x$:
$-5x - 5 \leq 15$
The objective is to solve the linear inequality \(-5x - 5 \leq 15\) for \(x\). This involves isolating \(x\) on one side of the inequality. We must be mindful of the rules for manipulating inequalities, particularly the rule that multiplying or dividing by a negative number reverses the direction of the inequality sign.
To begin, add \(5\) to both sides of the inequality to consolidate the constant terms:
\[-5x - 5 + 5 \leq 15 + 5\]
\[-5x \leq 20\]
This step simplifies the expression by isolating the term containing \(x\).
Now, divide both sides of the inequality by \(-5\). Because we are dividing by a negative number, the inequality sign must be reversed from \(\leq\) to \(\geq\):
\[x \geq \frac{20}{-5}\]
\[x \geq -4\]
This shows that all values of \(x\) greater than or equal to \(-4\) satisfy the original inequality. Comparing this result with the provided options, we see that it corresponds to the first option: \(x \geq -4\).
Linear inequalities are essential in budgeting and resource management. Suppose you have a budget of \(\$15\) to spend on a monthly subscription service that costs \(\$5\) per month, plus an initial setup fee of \(\$5\). If \(x\) represents the number of months you can afford, the cost is represented by the expression \(5x + 5 \leq 15\). This simple model helps in planning and making informed financial decisions, ensuring your expenses remain within your limits. In our case, the negative coefficient represents a situation where a value is decreasing over time.