Sure, let's solve the equation step-by-step:
[tex]\[
\frac{3x}{4} + 4x = \frac{7}{8} + 6x - 6
\][/tex]
First, let's combine like terms on both sides.
Combine the terms involving [tex]\(x\)[/tex] on the left side:
[tex]\[
\frac{3x}{4} + 4x = \frac{3x}{4} + \frac{16x}{4} = \frac{19x}{4}
\][/tex]
So, we have:
[tex]\[
\frac{19x}{4} = \frac{7}{8} + 6x - 6
\][/tex]
To make things easier, let's move everything involving [tex]\(x\)[/tex] to one side and constants to the other side. First, subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
\frac{19x}{4} - 6x = \frac{7}{8} - 6
\][/tex]
Now, convert [tex]\(6x\)[/tex] to a fraction with the same denominator as [tex]\(\frac{19x}{4}\)[/tex]:
[tex]\[
6x = \frac{24x}{4}
\][/tex]
Now the equation becomes:
[tex]\[
\frac{19x}{4} - \frac{24x}{4} = \frac{7}{8} - 6
\][/tex]
Combine the [tex]\(x\)[/tex] terms on the left side:
[tex]\[
\frac{19x - 24x}{4} = \frac{7}{8} - 6
\][/tex]
Simplify:
[tex]\[
-\frac{5x}{4} = \frac{7}{8} - 6
\][/tex]
To simplify further, we need to convert the constant [tex]\(6\)[/tex] to a fraction with the same denominator as [tex]\(\frac{7}{8}\)[/tex]:
[tex]\[
6 = \frac{48}{8}
\][/tex]
Now, substitute it into the equation:
[tex]\[
-\frac{5x}{4} = \frac{7}{8} - \frac{48}{8}
\][/tex]
Subtract the fractions on the right side:
[tex]\[
-\frac{5x}{4} = \frac{7 - 48}{8}
\][/tex]
Simplify the fraction:
[tex]\[
-\frac{5x}{4} = \frac{-41}{8}
\][/tex]
To isolate [tex]\(x\)[/tex], multiply both sides by [tex]\(-4/5\)[/tex]:
[tex]\[
x = \left(\frac{-41}{8}\right) \times \left(\frac{-4}{5}\right)
\][/tex]
First, multiply the numerators and the denominators:
[tex]\[
x = \frac{164}{40}
\][/tex]
Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:
[tex]\[
x = \frac{41}{10}
\][/tex]
So the solution is:
[tex]\[
x = 4.1
\][/tex]
Hence, the value of [tex]\(x\)[/tex] is [tex]\(4.1\)[/tex].
