Solve the inequality.
$2(4 x-3) \geq-3(3 x)+5 x$
We are given the linear inequality:
\[2(4x - 3) \geq -3(3x) + 5x\]
Our objective is to solve for \(x\) and find which of the provided options matches the solution set. We observe that this involves algebraic manipulation of a linear inequality. We will expand, simplify, and isolate the variable \(x\).
First, we expand the expressions on both sides of the inequality:
Now, substitute these back into the original inequality:
\[8x - 6 \geq -4x\]
Next, we isolate the variable \(x\) on one side of the inequality. We add \(4x\) to both sides:
\[8x + 4x - 6 \geq -4x + 4x\]
\[12x - 6 \geq 0\]
Then, add 6 to both sides:
\[12x \geq 6\]
Finally, divide both sides by 12:
\[x \geq \frac{6}{12}\]
\[x \geq 0.5\]
Comparing our derived solution \(x \geq 0.5\) with the given options:
The solution matches the first option. Thus, the solution to the inequality is \(x \geq 0.5\).
Linear inequalities are essential in real-life optimization problems. Imagine you are running a business and need to make a profit. Suppose your revenue function is \(R(x) = 8x\) and your cost function is \(C(x) = 6 - 4x\). To find the minimum production level \(x\) needed to ensure that revenue is at least equal to costs, you would set up the inequality \(8x \geq 6 - 4x\), which is exactly the type of problem we just solved. Understanding how to solve these inequalities helps in making informed decisions about production targets, budget management, and resource allocation.