Let's match the pairs of equivalent expressions step-by-step:
Step 1: Expand and simplify the expressions on the left.
1. [tex]\(\left(-14+\frac{3}{2} b\right)-\left(1+\frac{8}{2} b\right)\)[/tex]
Simplify:
[tex]\[
-14 + \frac{3}{2}b - 1 - 4b = -15 - \frac{5}{2}b
\][/tex]
2. [tex]\((5+2b)+\left(2b+\frac{3}{2}\right)\)[/tex]
Simplify:
[tex]\[
5 + 2b + 2b + \frac{3}{2} = 4b + \frac{13}{2}
\][/tex]
3. [tex]\(\left(\frac{7}{2} b-3\right)-(8+6b)\)[/tex]
Simplify:
[tex]\[
\frac{7}{2}b - 3 - 8 - 6b = \frac{7}{2}b - 6b - 3 - 8 = -\frac{5}{2}b - 11
\][/tex]
4. [tex]\((-10+b)+(7b-5)\)[/tex]
Simplify:
[tex]\[
-10 + b + 7b - 5 = 8b - 15
\][/tex]
Step 2: Identify the matching pairs based on the simplified expressions.
- [tex]\(\left(-14+\frac{3}{2} b\right)-\left(1+\frac{8}{2} b\right) \longleftrightarrow -15-\frac{5}{2}b\)[/tex]
- [tex]\((5+2b)+\left(2b+\frac{3}{2}\right) \longleftrightarrow 4b+\frac{13}{2}\)[/tex]
- [tex]\(\left(\frac{7}{2} b-3\right)-(8+6b) \longleftrightarrow -\frac{5}{2}b-11\)[/tex]
- [tex]\((-10+b)+(7b-5) \longleftrightarrow 8b-15\)[/tex]
So, the correct pairs are:
[tex]\[
\left(-14+\frac{3}{2} b\right)-\left(1+\frac{8}{2} b\right) \longleftrightarrow -15 - \frac{5}{2} b
\][/tex]
[tex]\[
(5+2 b) + \left(2 b+\frac{3}{2}\right) \longleftrightarrow 4 b+\frac{13}{2}
\][/tex]
[tex]\[
\left(\frac{7}{2} b-3\right)-(8+6 b) \longleftrightarrow \frac{-5}{2} b-11
\][/tex]
[tex]\[
(-10+b)+(7 b-5) \longleftrightarrow 8 b-15
\][/tex]
