To match the pairs of equivalent expressions, follow these detailed steps to simplify each expression:
1. Simplify [tex]\(\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \)[/tex]:
[tex]\[
(-14) + \frac{3}{2} b - 1 - 4b = -14 - 1 + \frac{3}{2} b - 4b = -15 - \frac{5}{2} b
\][/tex]
2. Simplify [tex]\(4b + \frac{13}{2}\)[/tex]:
[tex]\[
\text{This expression is already simplified.}
\][/tex]
3. Simplify [tex]\((5 + 2b) + \left(2b + \frac{3}{2}\right)\)[/tex]:
[tex]\[
5 + 2b + 2b + \frac{3}{2} = 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2}
\][/tex]
4. Simplify [tex]\(8b - 15\)[/tex]:
[tex]\[
\text{This expression is already simplified.}
\][/tex]
5. Simplify [tex]\(\left(\frac{7}{2}b - 3\right) - (8 + 6b)\)[/tex]:
[tex]\[
\frac{7}{2}b - 3 - 8 - 6b = \frac{7}{2}b - 6b - 11 = \frac{7}{2}b - \frac{12}{2}b - 11 = \frac{-5}{2}b - 11
\][/tex]
6. Simplify [tex]\(\frac{-5}{2}b - 11\)[/tex]:
[tex]\[
\text{This expression is already simplified.}
\][/tex]
7. Simplify [tex]\((-10 + b) + (7b - 5)\)[/tex]:
[tex]\[
-10 + b + 7b - 5 = 8b - 15
\][/tex]
8. Simplify [tex]\(-15 - \frac{5}{2}b\)[/tex]:
[tex]\[
\text{This expression is already simplified.}
\][/tex]
Now, match the equivalent expressions:
- [tex]\(\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \equiv -15 - \frac{5}{2}b\)[/tex]
- [tex]\(4b + \frac{13}{2} \equiv (5 + 2b) + \left(2b + \frac{3}{2}\right)\)[/tex]
- [tex]\(\left(\frac{7}{2}b - 3\right) - (8 + 6b) \equiv \frac{-5}{2}b - 11\)[/tex]
- [tex]\((-10 + b) + (7b - 5) \equiv 8b - 15\)[/tex]
So, the pairs are:
1. [tex]\(\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right)\)[/tex] with [tex]\(-15 - \frac{5}{2} b\)[/tex]
2. [tex]\(4 b + \frac{13}{2}\)[/tex] with [tex]\((5 + 2 b) + \left(2 b + \frac{3}{2}\right)\)[/tex]
3. [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6 b)\)[/tex] with [tex]\(\frac{-5}{2} b - 11\)[/tex]
4. [tex]\((-10 + b) + (7 b - 5)\)[/tex] with [tex]\(8 b - 15\)[/tex]
