Let's determine the correct pairs of equivalent expressions step-by-step.
1. Expression 1:
[tex]\[
\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right)
\][/tex]
Simplifying the components inside the parentheses:
[tex]\[
= -14 + \frac{3}{2} b - 1 - 4b
\][/tex]
Combine like terms:
[tex]\[
= -15 + \frac{3}{2} b - 4b = -15 - \frac{5}{2} b
\][/tex]
2. Expression 2:
[tex]\[
4 b + \frac{13}{2}
\][/tex]
This expression is in its simplest form.
3. Expression 3:
[tex]\[
(5 + 2 b) + \left(2 b + \frac{3}{2}\right)
\][/tex]
Simplifying the components inside the parentheses:
[tex]\[
= 5 + 2b + 2b + \frac{3}{2}
\][/tex]
Combine like terms:
[tex]\[
= 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2}
\][/tex]
4. Expression 4:
[tex]\[
8b - 15
\][/tex]
This expression is in its simplest form.
5. Expression 5:
[tex]\[
\left(\frac{7}{2} b - 3\right) - (8 + 6b)
\][/tex]
Simplifying the components inside the parentheses:
[tex]\[
= \frac{7}{2} b - 3 - 8 - 6b
\][/tex]
Combine like terms:
[tex]\[
= \frac{7}{2} b - 6b - 11 = -11 - \frac{5}{2} b
\][/tex]
6. Expression 6:
[tex]\[
\frac{-5}{2} b - 11
\][/tex]
This expression is in its simplest form.
7. Expression 7:
[tex]\[
(-10 + b) + (7b - 5)
\][/tex]
Simplifying the components inside the parentheses:
[tex]\[
= -10 + b + 7b - 5
\][/tex]
Combine like terms:
[tex]\[
= 8b - 15
\][/tex]
8. Expression 8:
[tex]\[
-15 - \frac{5}{2} b
\][/tex]
This expression is in its simplest form and equivalent to Expression 1.
Now, let's match the equivalent pairs:
1. [tex]\[
\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b
\][/tex]
2. [tex]\[
(5 + 2 b) + \left(2 b + \frac{3}{2}\right) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2}
\][/tex]
3. [tex]\[
\left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11
\][/tex]
4. [tex]\[
(-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15
\][/tex]
Thus, the pairs of equivalent expressions are:
[tex]\[
\begin{aligned}
&(-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b, \\
&(5 + 2 b) + (2 b + \frac{3}{2}) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2}, \\
&\left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11, \\
&(-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15.
\end{aligned}
\][/tex]
