Verified Answer :

  • Distribute the constant \(\frac{1}{2}\) across the terms \(-12x\) and \(16\).
  • Calculate each resulting term: \(\frac{1}{2} \times (-12x) = -6x\) and \(\frac{1}{2} \times 16 = 8\).
  • Combine the results to get the final simplified expression \(-6x + 8\).
  • The simplified form is \(\boxed{-6x + 8}\).

Explanation

  1. Analysis of the Expression

The given expression to simplify is \(\frac{1}{2}(-12x + 16)\). This is a linear expression in the form of a constant multiplied by a binomial. Our goal is to perform the distribution of the factor \(\frac{1}{2}\) across the terms inside the parentheses to arrive at a simpler form.

  1. Applying the Distributive Property

To simplify the expression \(\frac{1}{2}(-12x + 16)\), we apply the distributive property of multiplication over addition (or subtraction), which states that \(a(b + c) = ab + ac\). Here, \(a = \frac{1}{2}\), \(b = -12x\), and \(c = 16\). Applying this, we distribute \(\frac{1}{2}\) to each term:
\(\frac{1}{2} \cdot (-12x) + \frac{1}{2} \cdot 16\).

  1. Performing Calculations and Final Simplification

Now we perform the individual multiplications:

  • For the first term: \(\frac{1}{2} \cdot (-12x) = -6x\).
  • For the second term: \(\frac{1}{2} \cdot 16 = 8\).

Combining these results, we get \(-6x + 8\). Thus, the simplified expression is \(-6x + 8\).

Examples

Imagine you are running a small shop and you have a promotional discount where you reduce the price of an item by half. If the price formula for an item was given by \(P(x) = -12x + 16\) dollars (depending on some variable \(x\), like weight or quantity), applying your discount means your new price formula becomes \(\frac{1}{2}(-12x + 16)\). By simplifying this, you find that the new price function is \(-6x + 8\), making it much easier to calculate the price for any given value of \(x\). This is a practical example of how algebraic simplification helps in managing business calculations efficiently.