Let's solve the equation step by step. The given equation is:
[tex]\[18\left[\frac{4x + 3x - 6}{1} = \frac{8x}{1} + \frac{5(2x - 4)}{18}\right].\][/tex]
Simplify inside the brackets first:
[tex]\[ \frac{4x + 3x - 6}{1} = 4x + 3x - 6 = 7x - 6. \][/tex]
Next, look at the right-hand side of the equation:
[tex]\[ \frac{8x}{1} = 8x. \][/tex]
And simplify the fraction:
[tex]\[ \frac{5(2x - 4)}{18}. \][/tex]
First, distribute the 5 across [tex]\(2x - 4\)[/tex]:
[tex]\[ 5(2x - 4) = 10x - 20. \][/tex]
Then, we have:
[tex]\[ \frac{10x - 20}{18}. \][/tex]
We can simplify [tex]\(\frac{10x - 20}{18}\)[/tex] further. Factor out a 2 from the numerator:
[tex]\[ \frac{10x - 20}{18} = \frac{2(5x - 10)}{18} = \frac{5x - 10}{9}. \][/tex]
Now substitute back into the equation:
[tex]\[ 18\left[7x - 6 = 8x + \frac{5x - 10}{9}\right]. \][/tex]
To clear the fraction, multiply every term by 18:
[tex]\[ 18(7x - 6) = 18\left(8x + \frac{5x - 10}{9}\right). \][/tex]
Distribute the 18:
[tex]\[ 18(7x - 6) = 126x - 108. \][/tex]
And for the right-hand side:
[tex]\[ 18 \left(8x + \frac{5x - 10}{9}\right) = 18 \cdot 8x + 18 \cdot \frac{5x - 10}{9}. \][/tex]
Simplify inside the parentheses:
[tex]\[ 18 \cdot 8x = 144x, \][/tex]
and
[tex]\[ 18 \cdot \frac{5x - 10}{9} = 2(5x - 10) = 10x - 20. \][/tex]
So the equation becomes:
[tex]\[ 126x - 108 = 144x + 10x - 20. \][/tex]
Simplify the right-hand side:
[tex]\[ 126x - 108 = 154x - 20. \][/tex]
Move all terms involving [tex]\(x\)[/tex] on one side and constants on the other side:
[tex]\[ 126x - 154x = -20 + 108, \][/tex]
[tex]\[ -28x = 88. \][/tex]
Finally, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{88}{-28} = -\frac{22}{7}. \][/tex]
So the solution to the given equation is:
[tex]\[ x = -\frac{22}{7}. \][/tex]
