Sure, let's simplify the expression step-by-step:
[tex]\[ \frac{4x - 6}{8} + \frac{6 - x}{3} \][/tex]
First, let's rewrite each term to make it easier to combine them.
1. Simplify the fraction \(\frac{4x - 6}{8}\):
[tex]\[
\frac{4x - 6}{8} = \frac{4(x - \frac{3}{2})}{8} = \frac{4x - 6}{8} = \frac{x - \frac{3}{2}}{2} = \frac{4x}{8} - \frac{6}{8} = \frac{x}{2} - \frac{3}{4}
\][/tex]
1. Simplify the second fraction \(\frac{6 - x}{3}\):
[tex]\[
\frac{6 - x}{3} = \frac{6}{3} - \frac{x}{3} = 2 - \frac{x}{3}
\][/tex]
1. Combine both simplified terms:
[tex]\[
\frac{x}{2} - \frac{3}{4} + 2 - \frac{x}{3}
\][/tex]
1. Find a common denominator for the fractions:
The common denominator for 2, 3, and 4 is 12, so we rewrite each fraction with a common denominator:
[tex]\[
\begin{align*}
\frac{x}{2} & = \frac{6x}{12}, \\
-\frac{3}{4} & = -\frac{9}{12}, \\
2 & = \frac{24}{12}, \\
-\frac{x}{3} & = -\frac{4x}{12}
\end{align*}
\][/tex]
1. Add up all the terms over the common denominator:
[tex]\[
\frac{6x}{12} - \frac{9}{12} + \frac{24}{12} - \frac{4x}{12}
\][/tex]
1. Combine the numerators over the common denominator:
[tex]\[
\frac{6x - 9 + 24 - 4x}{12} = \frac{2x + 15}{12}
\][/tex]
1. Separate constants and coefficients of \(x\):
[tex]\[
\frac{2x}{12} + \frac{15}{12} = \frac{x}{6} + \frac{5}{4}
\][/tex]
So, the simplified expression is:
[tex]\[ \frac{x}{6} + \frac{5}{4} \][/tex]
