What is the value of $x$ in the equation $2.5(6 x-4)=10+4(1.5+0.5 x)$ ?
The problem is to solve the linear equation \(2.5(6x - 4) = 10 + 4(1.5 + 0.5x)\) for the variable \(x\). This equation involves decimals and requires algebraic distribution and simplification to isolate \(x\). Our objective is to determine which of the given options matches the calculated value of \(x\).
To simplify the left side, we distribute the \(2.5\):
\[2.5 \times 6x - 2.5 \times 4 = 15x - 10.\]
To simplify the right side, we distribute the \(4\):
\[10 + 4 \times 1.5 + 4 \times 0.5x = 10 + 6 + 2x = 16 + 2x.\]
Now, we equate the two simplified sides:
\[15x - 10 = 16 + 2x.\]
We move the variable terms to the left and constant terms to the right:
\[15x - 2x = 16 + 10.\]
Combining like terms:
\[13x = 26.\]
Dividing both sides by 13:
\[x = \frac{26}{13} = 2.\]
The value of \(x\) is 2, which corresponds to the third choice provided.
Solving equations is essential in real-world scenarios, such as budgeting or resource allocation. Imagine you are comparing two service plans: one charges an initial fee plus a rate per unit of usage, while another uses a different fee structure. By setting their total costs equal and solving for the unknown 'usage' (\(x\)), you can determine the exact point where both plans cost the same, helping you make an informed decision based on your expected consumption.