Which graph represents the solution to the inequality?
$-2.4(x-6) \geq 52.8$
The problem asks us to find the solution to the linear inequality \(-2.4(x - 6) \geq 52.8\). A linear inequality is an expression that involves a variable (in this case, \(x\)) raised to the power of 1, and uses an inequality symbol such as \(\geq\), \(\leq\), \(>\), or \(<\). Our goal is to isolate \(x\) on one side of the inequality. When manipulating inequalities, we must remember that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
To isolate \(x\), we first distribute the \(-2.4\) to the terms inside the parentheses. Multiplying \(-2.4\) by \(x\) gives \(-2.4x\), and multiplying \(-2.4\) by \(-6\) results in \(14.4\). Thus, the inequality becomes:
\[-2.4x + 14.4 \geq 52.8\]
Next, we subtract \(14.4\) from both sides of the inequality:
\[-2.4x \geq 52.8 - 14.4\]
\[-2.4x \geq 38.4\]
Now, to isolate \(x\), we divide both sides by \(-2.4\). Because we are dividing by a negative number, we must flip the direction of the inequality symbol:
\[x \leq \frac{38.4}{-2.4}\]
\[x \leq -16\]
The solution to the inequality is \(x \leq -16\). On a number line, this represents all values of \(x\) that are less than or equal to \(-16\). This is represented by a solid circle at \(-16\) (indicating that \(-16\) is included in the solution set) and a ray extending to the left, which covers all numbers smaller than \(-16\).
Inequalities like this are useful when planning budgets with limits. For example, if you have a base budget for a project and you need to spend less than a certain amount on materials, you can express your constraints as an inequality. Solving for the variable helps determine the maximum quantity you can purchase while remaining within your financial threshold. Understanding these operations is essential for making informed decisions in real-life resource management.