11.5.7 Check Your Understanding Compare each pair of expressions. Complete the statements. $$ 2 x $$ and $$ 2+2 $$ are not $$ \vee $$ equivalent expressions. $$ x+x+x+x+x+x $$ and $$ 6 x $$ are equivalent expressions. $$ \frac{1}{2} x $$ and $$ \frac{x}{2} $$ are equivalent expressions.

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Let's analyze each pair of expressions step by step to determine their equivalence.

1. **Expressions: $$ 2x $$ and $$ 2 + 2 $$**

- The expression $$ 2x $$ represents a variable $$ x $$ multiplied by 2.
   - The expression $$ 2 + 2 $$ simplifies to 4, which is a constant.

Since one expression is variable-dependent and the other is a constant, they are not equivalent.

**Conclusion:** $$ 2x $$ and $$ 2 + 2 $$ are not equivalent expressions.

2. **Expressions: $$ x + x + x + x + x + x $$ and $$ 6x $$**

- The expression $$ x + x + x + x + x + x $$ can be simplified by counting the number of $$ x $$ terms, which is 6. Thus, it simplifies to $$ 6x $$.
   - The expression $$ 6x $$ is already in its simplest form.

Since both expressions simplify to the same form, they are equivalent.

**Conclusion:** $$ x + x + x + x + x + x $$ and $$ 6x $$ are equivalent expressions.

3. **Expressions: $$ \frac{1}{2}x $$ and $$ \frac{x}{2} $$**

- The expression $$ \frac{1}{2}x $$ means one-half of $$ x $$.
   - The expression $$ \frac{x}{2} $$ also means $$ x $$ divided by 2, which is the same as one-half of $$ x $$.

Since both expressions represent the same quantity, they are equivalent.

**Conclusion:** $$ \frac{1}{2}x $$ and $$ \frac{x}{2} $$ are equivalent expressions.

### Summary of Statements:
- $$ 2x $$ and $$ 2 + 2 $$ are not equivalent expressions.
- $$ x + x + x + x + x + x $$ and $$ 6x $$ are equivalent expressions.
- $$ \frac{1}{2}x $$ and $$ \frac{x}{2} $$ are equivalent expressions.
- $$ 2x $$ and $$ 2 + 2 $$ are not equivalent. - $$ x + x + x + x + x + x $$ and $$ 6x $$ are equivalent. - $$ \frac{1}{2}x $$ and $$ \frac{x}{2} $$ are equivalent.

11.5.7 Check Your Understanding Compare each pair of expressions. Complete the statements. \( 2 x \) and \( 2+2 \) are not \( \vee \) equivalent expressions. \( x+x+x+x+x+x \) and \( 6 x \) are equivalent expressions. \( \frac{1}{2} x \) and \( \frac{x}{2} \) are equivalent expressions.

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