Question

identify the error in generating an expression equivalent to $4 + 2x - \frac{1}{2}(10 - 4x)$. then correct the error.
$4 + 2x + \frac{-1}{2}(10 + -4x)$
$4 + 2x + -5 + 2x$
$4 + 2x - 5 + 2x$

Explanation:

Step1: Identify the distribution error

The original step incorrectly distributed $\frac{1}{2}$ to $10-4x$ as $-5 + -2x$ (written as $-5 + 2x$ in the wrong work). The correct distribution is:
$\frac{1}{2}(10 - 4x) = \frac{1}{2}(10) + \frac{1}{2}(-4x) = 5 - 2x$

Step2: Rewrite the full expression

Substitute the correct distributed term back into the original expression:
$4 + 2x - (5 - 2x)$

Step3: Distribute the negative sign

Apply the negative sign to both terms inside the parentheses:
$4 + 2x - 5 + 2x$

Step4: Combine like terms

Group constant terms and variable terms separately:
$(4 - 5) + (2x + 2x) = -1 + 4x$

Answer:

The error was an incorrect distribution of $\frac{1}{2}(10-4x)$ (the work incorrectly calculated this as $-5 + 2x$ instead of $5 - 2x$, and also misapplied the leading negative sign). The correct simplified expression is $\boldsymbol{4x - 1}$ (or $-1 + 4x$).