Question

Fill in the blanks!

Are these expressions equivalent?
-2 (4x+2/3) and -8x + 4/3

Check using substitution.
x=1

-2 (4(1)+2/3)
-2 (_+2/3)
-2 (12/3 + 2/3)
-2 (_/3) = -28/3

-8 (_) + 4/3
-8 + 4/3
-24/3 + 4/3
-_/3

So, these expressions are not equivalent.
Are these expressions equivalent?
Check by expanding the first expression.

-2 (4x + 2/3) and -8x + 4/3
-2 (_) + (-2) (2/3)
-8x - 4/_

These two expressions are not equivalent.

Answer 1

Let's fill in the blanks correctly:

### First Expression:
-2 (4x + 2/3)

When substituting \(x = 1\):
-2 (4(1) + 2/3)
-2 (4 + 2/3)
-2 (12/3 + 2/3)
-2 (14/3)

Calculating this gives:
\(-2 * \frac{14}{3} = -\frac{28}{3}\)

### Second Expression:
-8x + 4/3

When substituting \(x = 1\):
-8(1) + 4/3
-8 + 4/3
\(-24/3 + 4/3\)
\(-\frac{24}{3} + \frac{4}{3} = -\frac{20}{3}\)

Now we compare:
- The first expression gives us \(-\frac{28}{3}\).
- The second expression gives us \(-\frac{20}{3}\).

So, the expressions are **not equivalent**.

### Second Check - Expanding the First Expression:
Let's re-check:
-2 (4x + 2/3) expands to:
-2 * 4x + (-2) * (2/3)
-8x - 4/3

Now we compare this with:
-8x + 4/3 which we know does not match because:
-8x - 4/3 is not the same as -8x + 4/3.

Therefore, the conclusion is:
These two expressions **are not equivalent**.

### Final Result:
- The first expression evaluates to \(-\frac{28}{3}\).
- The second expression evaluates to \(-\frac{20}{3}\).
- Upon expansion, \(-2 (4x + 2/3)\) evaluates to \(-8x - 4/3\), confirming the expressions are not equivalent.

So, indeed, **the expressions are not equivalent**.