Solve the following inequality for $x$.
$-5 x-5 \leq 15$
The goal is to solve the linear inequality \(-5x - 5 \leq 15\) for the variable \(x\). The standard approach to solving linear inequalities involves isolating the variable on one side while ensuring that we correctly apply the algebraic rules governing inequalities. Specifically, we must be careful when multiplying or dividing by negative numbers, as this reverses the direction of the inequality.
To begin isolating the variable \(x\), we first add \(5\) to both sides of the inequality:
\[-5x - 5 + 5 \leq 15 + 5\]
Simplifying both sides gives:
\[-5x \leq 20\]
Next, we divide both sides by \(-5\) to solve for \(x\). Since we are dividing by a negative number, the inequality sign must be reversed from \(\leq\) to \(\geq\):
\[x \geq \frac{20}{-5}\]
\[x \geq -4\]
The solution to the inequality \(-5x - 5 \leq 15\) is \(x \geq -4\). Comparing this with the provided options:
We find that the correct choice is \(x \geq -4\).
In real life, linear inequalities are often used for budgeting or setting limits. For example, if you have a budget of \$15 and you owe \$5 to a friend, and you need to determine the maximum or minimum amount you can spend per day on items priced at \$5 each, you could use an inequality. Here, if \(x\) represents the number of days or units of an item, the equation \(-5x - 5 \leq 15\) helps identify the threshold beyond which your spending would exceed your available resources. Understanding how to handle negative coefficients is crucial for correctly interpreting these financial boundaries.