To solve the equation for [tex]\( x \)[/tex] in terms of [tex]\( c \)[/tex]:
[tex]\[
\frac{2}{3}\left(c x + \frac{1}{2}\right) - \frac{1}{4} = \frac{5}{2}
\][/tex]
we need to follow these steps:
1. Distribute and clear fractions: Start by distributing [tex]\(\frac{2}{3}\)[/tex] to the terms inside the parentheses and then simplify the equation:
[tex]\[
\frac{2}{3} \cdot c x + \frac{2}{3} \cdot \frac{1}{2} - \frac{1}{4} = \frac{5}{2}
\][/tex]
which simplifies to:
[tex]\[
\frac{2}{3} c x + \frac{2}{6} - \frac{1}{4} = \frac{5}{2}
\][/tex]
Since [tex]\(\frac{2}{6} = \frac{1}{3}\)[/tex]:
[tex]\[
\frac{2}{3} c x + \frac{1}{3} - \frac{1}{4} = \frac{5}{2}
\][/tex]
2. Combine constants on one side of the equation: To do that, we should combine [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{2}{3} c x + \left(\frac{1}{3} - \frac{1}{4}\right) = \frac{5}{2}
\][/tex]
To subtract [tex]\(\frac{1}{4}\)[/tex] from [tex]\(\frac{1}{3}\)[/tex], we need a common denominator. The least common multiple of 3 and 4 is 12:
[tex]\[
\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}
\][/tex]
So,
[tex]\[
\frac{4}{12} - \frac{3}{12} = \frac{1}{12}
\][/tex]
Thus,
[tex]\[
\frac{2}{3} c x + \frac{1}{12} = \frac{5}{2}
\][/tex]
3. Isolate the term involving [tex]\( x \)[/tex]: Subtract [tex]\(\frac{1}{12}\)[/tex] from both sides to isolate the [tex]\(\frac{2}{3} c x\)[/tex] term:
[tex]\[
\frac{2}{3} c x = \frac{5}{2} - \frac{1}{12}
\][/tex]
Convert [tex]\(\frac{5}{2}\)[/tex] to have a denominator of 12:
[tex]\[
\frac{5}{2} = \frac{30}{12}
\][/tex]
[tex]\[
\frac{30}{12} - \frac{1}{12} = \frac{29}{12}
\][/tex]
Thus,
[tex]\[
\frac{2}{3} c x = \frac{29}{12}
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Multiply both sides of the equation by [tex]\(\frac{3}{2 c}\)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{29}{12} \cdot \frac{3}{2 c}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{29 \cdot 3}{12 \cdot 2 c} = \frac{87}{24 c}
\][/tex]
[tex]\[
x = \frac{29}{8 c}
\][/tex]
Therefore, the solution is:
[tex]\[
\boxed{x = \frac{29}{8 c}}
\][/tex]
