To match the pairs of equivalent expressions, let's first simplify each expression step by step.
1. Simplify \(\left(-14+\frac{3}{2} b\right)-\left(1+\frac{4}{b}\right)\):
[tex]\[
\left(-14 + \frac{3}{2} b\right) - \left(1 + 4b\right) = -14 + \frac{3}{2} b - 1 - 4b = -15 + \frac{3}{2}b - 4b = -15 - \frac{5}{2}b
\][/tex]
2. Simplify \(4b + \frac{13}{2}\):
[tex]\[
4b + \frac{13}{2}
\][/tex]
This expression is already simplified.
3. Simplify \((5 + 2b) + \left(2b + \frac{3}{2}\right)\):
[tex]\[
(5 + 2b) + (2b + \frac{3}{2}) = 5 + 2b + 2b + \frac{3}{2} = 5 + 4b + \frac{3}{2} = 5 + \frac{3}{2} + 4b = \frac{10}{2} + \frac{3}{2} + 4b = \frac{13}{2} + 4b = 4b + \frac{13}{2}
\][/tex]
4. Simplify \(\left(\frac{7}{2} b - 3 \right) - \left(8 + 6b\right)\):
[tex]\[
\left(\frac{7}{2} b - 3\right) - \left(8 + 6b\right) = \frac{7}{2} b - 3 - 8 - 6b = \frac{7}{2} b - 6b - 3 - 8 = \frac{7}{2} b - \frac{12}{2} b - 11 = -\frac{5}{2} b - 11
\][/tex]
5. Simplify \((-10 + b) + (7b - 5)\):
[tex]\[
(-10 + b) + (7b - 5) = -10 + b + 7b - 5 = -10 - 5 + b + 7b = -15 + 8b
\][/tex]
6. Simplify \(-15 - \frac{5}{2} b\):
[tex]\[
-15 - \frac{5}{2} b
\][/tex]
This expression is already simplified.
To find the pairs, we should find equivalent simplified expressions.
- From step 1 and step 6, we have: \( \left(-14 + \frac{3}{2} b\right)-\left(1 + 4b\right) = -15 - \frac{5}{2} b \)
- From step 2 and step 3, we have: \( 4b + \frac{13}{2} = (5 + 2b) + \left(2b + \frac{3}{2}\right) \)
- From step 4 and step 5, we have: \( \left(\frac{7}{2} b - 3\right)-\left(8 + 6b\right) = \frac{-5}{2} b - 11 \)
Thus, the correct pairs are:
[tex]\[
\left(-14 + \frac{3}{2} b\right) -\left( 1 + 4b \right) \longleftrightarrow -15 - \frac{5}{2} b
\][/tex]
[tex]\[
4b + \frac{13}{2} \longleftrightarrow (5 + 2b) + \left( 2b + \frac{3}{2} \right)
\][/tex]
[tex]\[
\left(\frac{7}{2} b - 3 \right) - (8 + 6b) \longleftrightarrow \frac{-5}{2} b -11
\][/tex]
