Question

Indicate whether each expression in the table is equivalent to 12x−1\frac{1}{2}x-1
2
1
​
x−1, is equivalent to x−12x-\frac{1}{2}x−
2
1
​
, or not equivalent to 12x−1\frac{1}{2}x-1
2
1
​
x−1 or x−12x-\frac{1}{2}x−
2
1
​
.
​Select all appropriate cells in the table.​
E
q
u
i
v
a
l
e
n
t

t
o

1
2
x
−
1
Equivalent to
2
1
​
x−1
E
q
u
i
v
a
l
e
n
t

t
o

x
−
1
2
Equivalent to x−
2
1
​

N
o
t

E
q
u
i
v
a
l
e
n
t

t
o

1
2
x
−
1

o
r

x
−
1
2
Not Equivalent to
2
1
​
x−1 or x−
2
1
​

2
3
(
3
4

x

−
3
2
)
3
2
​
(
4
3
​
x −
2
3
​
)




(
2
x
+
1
)

−

(
x
+
3
2
)
(2x+1) − (x+
2
3
​
)

Answer 1

We need to analyze whether each expression is equivalent to either \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \).

Let's evaluate the expressions given in your table.

### Given Expressions:

1. **Expression:** \( 2 \cdot \frac{3}{4} x - \frac{3}{2} \)
- Simplification:
\[
2 \cdot \frac{3}{4} x - \frac{3}{2} = \frac{3}{2} x - \frac{3}{2}
\]
- This does not simplify to \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \).
- Result: **Not Equivalent**

2. **Expression:** \( (2x + 1) - (x + \frac{2}{3}) \)
- Simplification:
\[
2x + 1 - x - \frac{2}{3} = x + (1 - \frac{2}{3}) = x + \frac{1}{3}
\]
- This does not simplify to \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \).
- Result: **Not Equivalent**

### Summary of Results:

- **Expression:** \( 2 \cdot \frac{3}{4} x - \frac{3}{2} \)
- Result: **Not Equivalent to \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \)**

- **Expression:** \( (2x + 1) - (x + \frac{2}{3}) \)
- Result: **Not Equivalent to \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \)**

Based on this evaluation, none of the expressions are equivalent to either \( 12x - \frac{1}{2}x - 1 \) or \( x - 12x - \frac{1}{2}x - 2 \). You can select "Not Equivalent" as the answer for both expressions in the table.