- Apply the distributive property to expand \(\frac{1}{4}(4x - 8)\) as \(x - 2\).
- Apply the distributive property to expand \(-2(5x + 6)\) as \(-10x - 12\).
- Combine the results: \((x - 2) - 10x - 12 = -9x - 14\).
- The simplest form is \(\boxed{-9x - 14}\), which corresponds to option b.
Explanation
- Problem Analysis
We are given the algebraic expression \(\frac{1}{4}(4x - 8) - 2(5x + 6)\). To simplify this expression, we will use the distributive property, which states that \(a(b + c) = ab + ac\). Our goal is to expand the terms, combine like terms, and find the simplest form from the provided choices.
- Expanding the Terms
First, we distribute the constants into the parentheses:
- For the first term \(\frac{1}{4}(4x - 8)\):
\(\frac{1}{4} \cdot 4x = x\)
\(\frac{1}{4} \cdot (-8) = -2\)
So, the first part simplifies to \(x - 2\).
- For the second term \(-2(5x + 6)\):
\(-2 \cdot 5x = -10x\)
\(-2 \cdot 6 = -12\)
So, the second part simplifies to \(-10x - 12\).
- Combining Like Terms and Conclusion
Now we combine the expanded parts:
\((x - 2) + (-10x - 12)\)
Rearrange to group the variable terms and the constant terms together:
\(x - 10x - 2 - 12\)
Combine the terms:
\(x - 10x = -9x\)
\(-2 - 12 = -14\)
Thus, the simplified form of the expression is \(-9x - 14\). Comparing this result with the provided options, we see that it matches option b.
Examples
Understanding the distributive property is crucial in real-life scenarios such as calculating total costs. For example, if you are buying 4 items priced at \(\$(x - 2)\) each and then discover a discount of \(\$2\) on each of the \(5x+6\) items you bought, you would use this algebraic method to determine the final cost or total change. It allows for efficient management of budgets and quick mental estimation in retail or resource planning.