Solve for $x$: $9x + 6 \geq 8x + 8$
The problem asks us to solve the linear inequality \(9x + 6 \geq 8x + 8\) for the variable \(x\). We have one variable \(x\) and a standard linear inequality involving addition and subtraction. Our objective is to isolate \(x\) on one side of the inequality symbol. To achieve this, we will use the property that adding or subtracting the same value from both sides of an inequality preserves its direction.
To isolate \(x\), we first subtract \(8x\) from both sides of the inequality:
\[9x - 8x + 6 \geq 8x - 8x + 8\]
Simplifying this expression gives us:
\[x + 6 \geq 8\]
Now, we need to isolate \(x\) by subtracting 6 from both sides of the inequality:
\[x + 6 - 6 \geq 8 - 6\]
This simplifies to:
\[x \geq 2\]
Comparing our result, \(x \geq 2\), with the given options, we identify that the correct choice is \(x \geq 2\). Therefore, the solution is \(\boxed{x \geq 2}\).
Linear inequalities are essential in real-world scenarios where constraints are present. For example, if you are planning a party and have a budget constraint, you might use an inequality to determine how many guests you can invite based on the cost per person. If the venue costs a flat fee plus a per-person charge, you would set up an inequality like \(Total Cost \leq Budget\) to find the maximum number of people you can accommodate while staying within your financial limits. Understanding these calculations helps in effective resource allocation and decision-making.