Solve the following inequality for $x$:
$-5 x-5 \leq 15$
We are asked to solve the linear inequality \(-5x - 5 \leq 15\) for the variable \(x\). Our goal is to isolate \(x\) on one side of the inequality. We can achieve this by using standard algebraic manipulations, being careful with the rules governing inequalities. Specifically, we know that adding or subtracting the same value from both sides does not change the inequality, while multiplying or dividing both sides by a negative number reverses the direction of the inequality.
To begin isolating \(x\), we first add 5 to both sides of the inequality:
\[-5x - 5 + 5 \leq 15 + 5\]
Simplifying both sides, we get:
\[-5x \leq 20\]
Now, we need to isolate \(x\) by dividing both sides by \(-5\). Because we are dividing by a negative number, the direction of the inequality must be reversed:
\[x \geq \frac{20}{-5}\]
Performing the division:
\[x \geq -4\]
We have found that the solution to the inequality is \(x \geq -4\). Comparing this result with the given choices:
The correct choice is \(x \geq -4\).
This type of linear inequality is fundamental in many real-life applications. For example, imagine you are managing a budget for a monthly subscription service that costs \$5 per month, and you also have an initial setup fee of \$5. If your total spending limit is \$15, you want to know how many months (represented by \(x\)) you can afford. The inequality \(-5x - 5 \geq -15\) (or rearranged as \(-5x - 5 \leq 15\)) helps you determine the maximum duration or bounds of your budget. Understanding how to solve these inequalities ensures you can make informed financial decisions and stay within your limits!