- Use the distributive property to expand the expression: \(2x(2+1) - 3(x+3) = 4x + 2x - 3x - 9\).
- Combine the like terms containing \(x\): \(4x + 2x - 3x = 3x\).
- Combine the constant terms to get the simplified result: \(3x - 9\).
- The final simplified expression is \(\boxed{3x-9}\).
Explanation
- Analysis of the Expression
The expression to be simplified is \(2x(2+1) - 3(x+3)\). To simplify this, we need to apply the distributive property, which states that \(a(b+c) = ab + ac\). We perform this operation on both parts of the expression independently.
- Distributing Terms
We first distribute \(2x\) into the parentheses \((2+1)\), resulting in \(2x(2) + 2x(1) = 4x + 2x\). Then we distribute \(-3\) into the parentheses \((x+3)\), resulting in \(-3(x) - 3(3) = -3x - 9\). Putting these together, we have the expanded expression:
\[4x + 2x - 3x - 9.\]
- Combining Like Terms and Final Result
Now we identify the like terms, which are the terms containing \(x\). These are \(4x\), \(2x\), and \(-3x\). Combining them:
\[4x + 2x - 3x = (4 + 2 - 3)x = 3x.\]
After combining the like terms, the constant term \(-9\) remains unchanged. The final expression is \(3x - 9\).
Examples
Understanding how to simplify expressions is fundamental in algebra. Imagine you are managing a small store's inventory where you have \(2x\) boxes, each containing \((2+1)\) items, and you need to remove \(3\) sets of \((x+3)\) items. Simplifying this helps determine exactly how many items you have left: \(2x(3) - 3(x+3) = 6x - 3x - 9 = 3x - 9\). This skill is essential for budgeting, managing resources, and solving real-world mathematical models.