Question

ndicate 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

To determine the equivalence of the expressions given, let's first break down the original expressions and analyze the new ones.

The two expressions we need to look for equivalence to are:
1. \( \frac{1}{2}x - 1 \)
2. \( x - \frac{1}{2} \)

Now, we will analyze each expression provided in your table.

1. **Expression: \( \frac{3}{2} \left( \frac{3}{4} x - \frac{3}{2} \right) \)**

Simplifying:
\[
= \frac{3}{2} \cdot \frac{3}{4}x - \frac{3}{2} \cdot \frac{3}{2}
= \frac{9}{8}x - \frac{9}{4}
\]
This does not simplify to either \( \frac{1}{2}x - 1 \) or \( x - \frac{1}{2} \) so it is **Not Equivalent**.

2. **Expression: \( (2x + 1) - (x + \frac{2}{3}) \)**

Simplifying:
\[
= 2x + 1 - x - \frac{2}{3}
= (2x - x) + (1 - \frac{2}{3})
= x + \frac{1}{3}
\]
This does not simplify to either \( \frac{1}{2}x - 1 \) or \( x - \frac{1}{2} \) so it is **Not Equivalent**.

Now to summarize the results:

- For \( \frac{3}{2} \left( \frac{3}{4} x - \frac{3}{2} \right) \): **Not Equivalent to** \( \frac{1}{2}x - 1 \) or \( x - \frac{1}{2} \).

- For \( (2x + 1) - (x + \frac{2}{3}) \): **Not Equivalent to** \( \frac{1}{2}x - 1 \) or \( x - \frac{1}{2} \).

Therefore, both expressions are **Not Equivalent** to either of the original expressions provided.