Let's solve the equation step by step:
[tex]\[
\frac{b-4}{3} - \frac{2b+1}{6} = \frac{5b-1}{2}
\][/tex]
First, we need to find a common denominator for the fractions. The denominators we have are 3, 6, and 2. The least common multiple of these numbers is 6. Hence, we will convert each term so that they have a denominator of 6.
Rewriting each term with a common denominator:
[tex]\[
\frac{b-4}{3} = \frac{2(b-4)}{6} = \frac{2b-8}{6}
\][/tex]
[tex]\[
\frac{2b+1}{6} = \frac{2b+1}{6}
\][/tex]
[tex]\[
\frac{5b-1}{2} = \frac{3(5b-1)}{6} = \frac{15b-3}{6}
\][/tex]
Now, substituting these back into the equation, we get:
[tex]\[
\frac{2b-8}{6} - \frac{2b+1}{6} = \frac{15b-3}{6}
\][/tex]
Since the denominators are the same, we can combine the numerators:
[tex]\[
\frac{(2b-8) - (2b+1)}{6} = \frac{15b-3}{6}
\][/tex]
Simplify the numerator on the left-hand side:
[tex]\[
\frac{2b - 8 - 2b - 1}{6} = \frac{15b-3}{6}
\][/tex]
[tex]\[
\frac{-9}{6} = \frac{15b-3}{6}
\][/tex]
Since the denominators are the same, we can equate the numerators directly:
[tex]\[
-9 = 15b - 3
\][/tex]
Now solve for [tex]\( b \)[/tex]. First, add 3 to both sides:
[tex]\[
-9 + 3 = 15b
\][/tex]
[tex]\[
-6 = 15b
\][/tex]
Next, divide by 15:
[tex]\[
b = \frac{-6}{15}
\][/tex]
Simplify the fraction:
[tex]\[
b = -\frac{2}{5}
\][/tex]
Thus, the solution to the equation [tex]\(\frac{b-4}{3} - \frac{2b+1}{6} = \frac{5b-1}{2}\)[/tex] is:
[tex]\[
b = -\frac{2}{5}
\][/tex]
