To solve the inequality
[tex]\[
2 \frac{3}{5}we need to isolate [tex]\( b \)[/tex].
Here is the detailed step-by-step solution:
1. First, convert the mixed number [tex]\( 2 \frac{3}{5} \)[/tex] into an improper fraction.
[tex]\[
2 \frac{3}{5} = 2 + \frac{3}{5} = \frac{2 \cdot 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}.
\][/tex]
2. Now, rewrite the inequality with the improper fractions:
[tex]\[
\frac{13}{5} < b - \frac{8}{15}.
\][/tex]
3. To isolate [tex]\( b \)[/tex], add [tex]\( \frac{8}{15} \)[/tex] to both sides of the inequality. To do this, we need to find a common denominator for the fractions [tex]\( \frac{13}{5} \)[/tex] and [tex]\( \frac{8}{15} \)[/tex].
The least common multiple of 5 and 15 is 15.
4. Convert [tex]\( \frac{13}{5} \)[/tex] to have a denominator of 15:
[tex]\[
\frac{13}{5} = \frac{13 \cdot 3}{5 \cdot 3} = \frac{39}{15}.
\][/tex]
5. Now add [tex]\( \frac{39}{15} \)[/tex] and [tex]\( \frac{8}{15} \)[/tex]:
[tex]\[
\frac{39}{15} + \frac{8}{15} = \frac{39 + 8}{15} = \frac{47}{15}.
\][/tex]
6. So the inequality now becomes:
[tex]\[
b > \frac{47}{15}.
\][/tex]
7. Convert [tex]\( \frac{47}{15} \)[/tex] back to a mixed number to get the final result:
[tex]\[
\frac{47}{15} = 3 \frac{2}{15}.
\][/tex]
Here, [tex]\( 47 \div 15 = 3 \)[/tex] with a remainder of [tex]\( 2 \)[/tex]. Thus:
[tex]\[
\frac{47}{15} = 3 \frac{2}{15}.
\][/tex]
Therefore, the final inequality is:
[tex]\[
b > 3 \frac{2}{15}.
\][/tex]
The correct answer is:
[tex]\[
b > 3 \frac{2}{15}.
\][/tex]
