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Sagot :
Sure, let's match the pairs of equivalent expressions step by step. Here is the detailed analysis of each expression:
1. [tex]\(\left(-14 + \frac{3}{2} b \right) - \left(1 + \frac{8}{2} b \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\( -14 + \frac{3}{2} b \)[/tex]
- [tex]\( -1 - 4b \)[/tex] (since [tex]\( \frac{8}{2} = 4 \)[/tex])
Combining the expressions:
- [tex]\( -14 - 1 + \frac{3}{2} b - 4b \)[/tex]
- [tex]\( -15 - \frac{5}{2} b \)[/tex]
This matches [tex]\(-15 - \frac{5}{2} b\)[/tex], so we pair it with the last tile:
[tex]\[ \boxed{-15 - \frac{5}{2}b} \][/tex]
2. [tex]\(4b + \frac{13}{2}\)[/tex]:
This cannot be simplified any further and remains:
- [tex]\(4b + \frac{13}{2}\)[/tex]
This matches directly with [tex]\(4b + \frac{13}{2}\)[/tex], so we pair it with the second tile:
[tex]\[ \boxed{(5 + 2b) + \left( 2b + \frac{3}{2} \right)} \][/tex]
3. [tex]\((5 + 2b) + \left( 2b + \frac{3}{2} \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\(5 + 2b\)[/tex]
- [tex]\(2b + \frac{3}{2}\)[/tex]
Combining the expressions:
- [tex]\(5 + 2b + 2b + \frac{3}{2}\)[/tex]
- [tex]\(4b + 5 + \frac{3}{2}\)[/tex]
- [tex]\(4b + \frac{10}{2} + \frac{3}{2}\)[/tex]
- [tex]\(4b + \frac{13}{2}\)[/tex]
This matches [tex]\(\left( 5 + 2b \right) + \left( 2b + \frac{3}{2} \right)\)[/tex], so we pair it with the fourth tile:
[tex]\[ \boxed{4b + \frac{13}{2}} \][/tex]
4. [tex]\(8b - 15\)[/tex]:
This is already simplified and matches:
- [tex]\(8b - 15\)[/tex], so we pair it directly with the third tile:
[tex]\[ \boxed{8b - 15} \][/tex]
5. [tex]\(\left( \frac{7}{2}b - 3 \right) - \left(8 + 6b \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\( \frac{7}{2}b - 3 \)[/tex]
- [tex]\( -8 - 6b \)[/tex]
Combining the expressions:
- [tex]\( \frac{7}{2}b - 3 - 8 - 6b \)[/tex]
- [tex]\( \frac{7}{2}b - 6b - 11 \)[/tex]
- [tex]\(\frac{7}{2}b - \frac{12}{2}b - 11\)[/tex]
- [tex]\(-\frac{5}{2}b - 11\)[/tex]
This matches [tex]\(\left( \frac{7}{2}b - 3 \right) - \left( 8 + 6b \right)\)[/tex], so we pair it directly with the first tile:
[tex]\[ \boxed{-\frac{5}{2}b - 11} \][/tex]
The completed pairs are:
- [tex]\(\left(-14 + \frac{3}{2} b \right) - \left(1 + \frac{8}{2} b\right) \quad \rightarrow \quad -15 - \frac{5}{2} b\)[/tex]
- [tex]\(4b + \frac{13}{2} \quad \rightarrow \quad (5 + 2b) + \left( 2b + \frac{3}{2} \right)\)[/tex]
- [tex]\(8b - 15 \quad \rightarrow \quad (-10 + b) + (7b - 5)\)[/tex]
- [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b) \quad \rightarrow \quad -\frac{5}{2} b - 11\)[/tex]
1. [tex]\(\left(-14 + \frac{3}{2} b \right) - \left(1 + \frac{8}{2} b \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\( -14 + \frac{3}{2} b \)[/tex]
- [tex]\( -1 - 4b \)[/tex] (since [tex]\( \frac{8}{2} = 4 \)[/tex])
Combining the expressions:
- [tex]\( -14 - 1 + \frac{3}{2} b - 4b \)[/tex]
- [tex]\( -15 - \frac{5}{2} b \)[/tex]
This matches [tex]\(-15 - \frac{5}{2} b\)[/tex], so we pair it with the last tile:
[tex]\[ \boxed{-15 - \frac{5}{2}b} \][/tex]
2. [tex]\(4b + \frac{13}{2}\)[/tex]:
This cannot be simplified any further and remains:
- [tex]\(4b + \frac{13}{2}\)[/tex]
This matches directly with [tex]\(4b + \frac{13}{2}\)[/tex], so we pair it with the second tile:
[tex]\[ \boxed{(5 + 2b) + \left( 2b + \frac{3}{2} \right)} \][/tex]
3. [tex]\((5 + 2b) + \left( 2b + \frac{3}{2} \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\(5 + 2b\)[/tex]
- [tex]\(2b + \frac{3}{2}\)[/tex]
Combining the expressions:
- [tex]\(5 + 2b + 2b + \frac{3}{2}\)[/tex]
- [tex]\(4b + 5 + \frac{3}{2}\)[/tex]
- [tex]\(4b + \frac{10}{2} + \frac{3}{2}\)[/tex]
- [tex]\(4b + \frac{13}{2}\)[/tex]
This matches [tex]\(\left( 5 + 2b \right) + \left( 2b + \frac{3}{2} \right)\)[/tex], so we pair it with the fourth tile:
[tex]\[ \boxed{4b + \frac{13}{2}} \][/tex]
4. [tex]\(8b - 15\)[/tex]:
This is already simplified and matches:
- [tex]\(8b - 15\)[/tex], so we pair it directly with the third tile:
[tex]\[ \boxed{8b - 15} \][/tex]
5. [tex]\(\left( \frac{7}{2}b - 3 \right) - \left(8 + 6b \right)\)[/tex]:
Simplifying the expression inside the parentheses:
- [tex]\( \frac{7}{2}b - 3 \)[/tex]
- [tex]\( -8 - 6b \)[/tex]
Combining the expressions:
- [tex]\( \frac{7}{2}b - 3 - 8 - 6b \)[/tex]
- [tex]\( \frac{7}{2}b - 6b - 11 \)[/tex]
- [tex]\(\frac{7}{2}b - \frac{12}{2}b - 11\)[/tex]
- [tex]\(-\frac{5}{2}b - 11\)[/tex]
This matches [tex]\(\left( \frac{7}{2}b - 3 \right) - \left( 8 + 6b \right)\)[/tex], so we pair it directly with the first tile:
[tex]\[ \boxed{-\frac{5}{2}b - 11} \][/tex]
The completed pairs are:
- [tex]\(\left(-14 + \frac{3}{2} b \right) - \left(1 + \frac{8}{2} b\right) \quad \rightarrow \quad -15 - \frac{5}{2} b\)[/tex]
- [tex]\(4b + \frac{13}{2} \quad \rightarrow \quad (5 + 2b) + \left( 2b + \frac{3}{2} \right)\)[/tex]
- [tex]\(8b - 15 \quad \rightarrow \quad (-10 + b) + (7b - 5)\)[/tex]
- [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b) \quad \rightarrow \quad -\frac{5}{2} b - 11\)[/tex]
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