What is the solution set for the following inequality? \( \begin{array}{l}6 x+12 \geq 2 x+10 \\ x \geq \geq 2 \\ x \leq 2 \\ x \leq-0.5\end{array} \)

To begin, let's simplify the inequality \(6x + 12 \geq 2x + 10\). Subtract \(2x\) from both sides: \[ 6x - 2x + 12 \geq 10 \] This simplifies to: \[ 4x + 12 \geq 10 \] Next, subtract 12 from both sides: \[ 4x \geq -2 \] Dividing both sides by 4 gives: \[ x \geq -0.5 \] Now let's analyze the solution set: 1. \(x \geq -0.5\) 2. \(x \geq 2\) 3. \(x \leq 2\) 4. \(x \leq -0.5\) From this analysis, we see that \(x \geq 2\) contradicts \(x \leq 2\) and any values \(x \leq -0.5\) contradict \(x \geq -0.5\). Thus, the solutions boil down to just \(x \geq -0.5\), where \(x\) can be greater than or equal to -0.5. However, since \(x\) contradicts other conditions, the final solution set is: **Final Solution Set:** \(x \geq 2\) (as it overshadows the other conditions).