Sure, let's solve the equation:
[tex]\[
\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6}
\][/tex]
### Step-by-Step Solution:
1. Combine the constants on the left side of the equation:
First, we need to combine the fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{7}{2}\)[/tex]. To do this, we need a common denominator, which in this case is 6.
[tex]\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\][/tex]
[tex]\[
\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}
\][/tex]
Adding these together:
[tex]\[
\frac{4}{6} + \frac{21}{6} = \frac{4 + 21}{6} = \frac{25}{6}
\][/tex]
So, our equation now looks like:
[tex]\[
\frac{25}{6} - 4x = -9x + \frac{5}{6}
\][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
Add [tex]\(9x\)[/tex] to both sides to move all [tex]\(x\)[/tex] terms to the left side:
[tex]\[
\frac{25}{6} - 4x + 9x = \frac{5}{6}
\][/tex]
Simplify the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{25}{6} + 5x = \frac{5}{6}
\][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]:
Subtract [tex]\(\frac{25}{6}\)[/tex] from both sides to move the constants to the right side:
[tex]\[
5x = \frac{5}{6} - \frac{25}{6}
\][/tex]
Combine the fractions on the right side:
[tex]\[
5x = \frac{5 - 25}{6} = \frac{-20}{6} = -\frac{10}{3}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-\frac{10}{3}}{5}
\][/tex]
Simplify the fraction:
[tex]\[
x = -\frac{10}{3} \times \frac{1}{5} = -\frac{10}{15} = -\frac{2}{3}
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = -\frac{2}{3}
\][/tex]
