To solve this problem, we need to pair equivalent algebraic expressions. Each of these expressions simplifies to a common form. Let's examine each:
1. [tex]\((-14+\frac{3}{2} b) - (1+\frac{8}{2} b)\)[/tex]:
Simplifies to:
[tex]\[
-14 + \frac{3}{2} b - 1 - 4b = -15 - \frac{5}{2} b
\][/tex]
2. [tex]\(4b + \frac{13}{2}\)[/tex]:
This expression is already simplified:
[tex]\[
4b + \frac{13}{2}
\][/tex]
3. [tex]\((5+2b) + (2b+\frac{3}{2})\)[/tex]:
Simplifies to:
[tex]\[
5 + 2b + 2b + \frac{3}{2} = \frac{10}{2} + 4b + \frac{3}{2} = 4b + \frac{13}{2}
\][/tex]
4. [tex]\(8b - 15\)[/tex]:
This expression is already simplified:
[tex]\[
8b - 15
\][/tex]
5. [tex]\((\frac{7}{2} b - 3) - (8 + 6b)\)[/tex]:
Simplifies to:
[tex]\[
\frac{7}{2} b - 3 - 8 - 6b = -11 - \frac{5}{2} b
\][/tex]
6. [tex]\(\frac{-5}{2} b - 11\)[/tex]:
This expression is already simplified:
[tex]\[
\frac{-5}{2} b - 11
\][/tex]
7. [tex]\((-10 + b) + (7b - 5)\)[/tex]:
Simplifies to:
[tex]\[
-10 + b + 7b - 5 = 8b - 15
\][/tex]
8. [tex]\(-15 - \frac{5}{2} b\)[/tex]:
This expression is already simplified:
[tex]\[
-15 - \frac{5}{2} b
\][/tex]
Now we can match the expressions that are equivalent:
1. [tex]\((-14+\frac{3}{2} b) - (1+\frac{8}{2} b) \leftrightarrow -15 - \frac{5}{2} b\)[/tex]
2. [tex]\(4b + \frac{13}{2} \leftrightarrow (5+2b) + (2b+\frac{3}{2})\)[/tex]
3. [tex]\(8b - 15 \leftrightarrow (-10+b) + (7b-5)\)[/tex]
4. [tex]\((\frac{7}{2} b - 3) - (8+6b) \leftrightarrow \frac{-5}{2} b - 11\)[/tex]
Thus, the correct pairs are:
[tex]\[
\begin{array}{c}
(-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \longleftrightarrow -15 - \frac{5}{2} b, \\
4b + \frac{13}{2} \longleftrightarrow (5+2b) + (2b+ \frac{3}{2}), \\
8b - 15 \longleftrightarrow (-10 + b) + (7b - 5), \\
(\frac{7}{2}b - 3) - (8 + 6b) \longleftrightarrow \frac{-5}{2}b - 11
\end{array}
\][/tex]
