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Let's determine the correct pairs of equivalent expressions step-by-step.
1. Expression 1:
[tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = -14 + \frac{3}{2} b - 1 - 4b \][/tex]
Combine like terms:
[tex]\[ = -15 + \frac{3}{2} b - 4b = -15 - \frac{5}{2} b \][/tex]
2. Expression 2:
[tex]\[ 4 b + \frac{13}{2} \][/tex]
This expression is in its simplest form.
3. Expression 3:
[tex]\[ (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = 5 + 2b + 2b + \frac{3}{2} \][/tex]
Combine like terms:
[tex]\[ = 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
4. Expression 4:
[tex]\[ 8b - 15 \][/tex]
This expression is in its simplest form.
5. Expression 5:
[tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6b) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = \frac{7}{2} b - 3 - 8 - 6b \][/tex]
Combine like terms:
[tex]\[ = \frac{7}{2} b - 6b - 11 = -11 - \frac{5}{2} b \][/tex]
6. Expression 6:
[tex]\[ \frac{-5}{2} b - 11 \][/tex]
This expression is in its simplest form.
7. Expression 7:
[tex]\[ (-10 + b) + (7b - 5) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = -10 + b + 7b - 5 \][/tex]
Combine like terms:
[tex]\[ = 8b - 15 \][/tex]
8. Expression 8:
[tex]\[ -15 - \frac{5}{2} b \][/tex]
This expression is in its simplest form and equivalent to Expression 1.
Now, let's match the equivalent pairs:
1. [tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b \][/tex]
2. [tex]\[ (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2} \][/tex]
3. [tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11 \][/tex]
4. [tex]\[ (-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15 \][/tex]
Thus, the pairs of equivalent expressions are:
[tex]\[ \begin{aligned} &(-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b, \\ &(5 + 2 b) + (2 b + \frac{3}{2}) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2}, \\ &\left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11, \\ &(-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15. \end{aligned} \][/tex]
1. Expression 1:
[tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = -14 + \frac{3}{2} b - 1 - 4b \][/tex]
Combine like terms:
[tex]\[ = -15 + \frac{3}{2} b - 4b = -15 - \frac{5}{2} b \][/tex]
2. Expression 2:
[tex]\[ 4 b + \frac{13}{2} \][/tex]
This expression is in its simplest form.
3. Expression 3:
[tex]\[ (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = 5 + 2b + 2b + \frac{3}{2} \][/tex]
Combine like terms:
[tex]\[ = 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
4. Expression 4:
[tex]\[ 8b - 15 \][/tex]
This expression is in its simplest form.
5. Expression 5:
[tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6b) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = \frac{7}{2} b - 3 - 8 - 6b \][/tex]
Combine like terms:
[tex]\[ = \frac{7}{2} b - 6b - 11 = -11 - \frac{5}{2} b \][/tex]
6. Expression 6:
[tex]\[ \frac{-5}{2} b - 11 \][/tex]
This expression is in its simplest form.
7. Expression 7:
[tex]\[ (-10 + b) + (7b - 5) \][/tex]
Simplifying the components inside the parentheses:
[tex]\[ = -10 + b + 7b - 5 \][/tex]
Combine like terms:
[tex]\[ = 8b - 15 \][/tex]
8. Expression 8:
[tex]\[ -15 - \frac{5}{2} b \][/tex]
This expression is in its simplest form and equivalent to Expression 1.
Now, let's match the equivalent pairs:
1. [tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b \][/tex]
2. [tex]\[ (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2} \][/tex]
3. [tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11 \][/tex]
4. [tex]\[ (-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15 \][/tex]
Thus, the pairs of equivalent expressions are:
[tex]\[ \begin{aligned} &(-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \xrightarrow{\longleftrightarrow} -15 - \frac{5}{2} b, \\ &(5 + 2 b) + (2 b + \frac{3}{2}) \xrightarrow{\longleftrightarrow} 4 b + \frac{13}{2}, \\ &\left(\frac{7}{2} b - 3\right) - (8 + 6 b) \xrightarrow{\longleftrightarrow} \frac{-5}{2} b - 11, \\ &(-10 + b) + (7 b - 5) \xrightarrow{\longleftrightarrow} 8 b - 15. \end{aligned} \][/tex]
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