Question

click to show whether each pair of expressions are equivalent.
equivalent
not equivalent
$\frac{1}{3}(4 - 2x) - 2$ and $-x$
$\frac{1}{2}(4 - 2x) + 2$ and $4 + x$

Explanation:

First Pair: $\boldsymbol{\frac{1}{3}(4 - 2x) - 2}$ and $\boldsymbol{-x}$

Step 1: Distribute $\frac{1}{3}$

Multiply $\frac{1}{3}$ with each term inside the parentheses: $\frac{1}{3} \times 4 - \frac{1}{3} \times 2x = \frac{4}{3} - \frac{2}{3}x$

Step 2: Subtract 2

Now subtract 2 from the result: $\frac{4}{3} - \frac{2}{3}x - 2$. Convert 2 to thirds: $2 = \frac{6}{3}$, so $\frac{4}{3} - \frac{6}{3} - \frac{2}{3}x = -\frac{2}{3} - \frac{2}{3}x$

Step 3: Compare with $-x$

The simplified form $-\frac{2}{3} - \frac{2}{3}x$ is not the same as $-x$ (since $-\frac{2}{3} - \frac{2}{3}x
eq -x$ for all $x$). So this pair is not equivalent.

Second Pair: $\boldsymbol{\frac{1}{2}(4 - 2x) + 2}$ and $\boldsymbol{4 + x}$

Step 1: Distribute $\frac{1}{2}$

Multiply $\frac{1}{2}$ with each term inside the parentheses: $\frac{1}{2} \times 4 - \frac{1}{2} \times 2x = 2 - x$

Step 2: Add 2

Now add 2 to the result: $2 - x + 2 = 4 - x$

Step 3: Compare with $4 + x$

The simplified form $4 - x$ is not the same as $4 + x$ (since $-x
eq x$ for $x
eq 0$). So this pair is not equivalent.

Answer:

• For $\frac{1}{3}(4 - 2x) - 2$ and $-x$: Not equivalent (check the "Not equivalent" box for this row).
• For $\frac{1}{2}(4 - 2x) + 2$ and $4 + x$: Not equivalent (check the "Not equivalent" box for this row).