Let's solve each expression step-by-step, matching them with their simplified forms:
1. Expression: \(\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right)\)
- Simplify \(\left(1 + \frac{8}{2} b\right)\): It becomes \(1 + 4b\).
- Now the expression is: \(-14 + \frac{3}{2} b - (1 + 4b)\).
- Distribute the negative sign: \(-14 + \frac{3}{2} b - 1 - 4b\).
- Combine like terms: \(-14 - 1 + \frac{3}{2} b - 4b = -15 - \frac{5}{2} b\).
- Simplified form: \(-\frac{5}{2} b - 15\).
2. Expression: \((5 + 2b) + \left(2b + \frac{3}{2}\right)\)
- Simplify: \(5 + 2b + 2b + \frac{3}{2}\).
- Combine like terms: \(5 + \frac{3}{2} + 4b\).
- Simplified form: \(4b + \frac{13}{2}\).
3. Expression: \(\left(\frac{7}{2} b - 3\right) - \left(8 + 6b\right)\)
- Simplify: \(\frac{7}{2} b - 3 - 8 - 6b\).
- Combine like terms: \(\frac{7}{2}b - 6b - 3 - 8 = -\frac{5}{2}b - 11\).
- Simplified form: \(-\frac{5}{2} b - 11\).
4. Expression: \((-10 + b) + (7b - 5)\)
- Simplify: \(-10 + b + 7b - 5\).
- Combine like terms: \(b + 7b - 10 - 5 = 8b - 15\).
- Simplified form: \(8b - 15\).
Therefore, the pairs of equivalent expressions are:
1. \(\left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right)\) matches with \(-\frac{5}{2} b - 15\)
2. \((5 + 2b) + \left(2b + \frac{3}{2}\right)\) matches with \(4b + \frac{13}{2}\)
3. \(\left(\frac{7}{2} b - 3\right) - (8 + 6b)\) matches with \(-\frac{5}{2} b - 11\)
4. [tex]\((-10 + b) + (7b - 5)\)[/tex] matches with [tex]\(8b - 15\)[/tex]
