To solve the equation \(\frac{3(5x + 4)}{9} + \frac{4(6x + 4)}{8} = 22\), let's go through the steps in detail:
1. Clear the fractions:
To eliminate the denominators, we find the least common multiple (LCM) of 9 and 8. The LCM of 9 and 8 is 72. We will multiply every term by 72 to clear the fractions.
[tex]\[
72 \cdot \left(\frac{3(5x + 4)}{9}\right) + 72 \cdot \left(\frac{4(6x + 4)}{8}\right) = 72 \cdot 22
\][/tex]
2. Simplify the equation:
We simplify each term by performing the multiplications:
[tex]\[
72 \cdot \left(\frac{3(5x + 4)}{9}\right) = 8 \cdot 3(5x + 4) = 24(5x + 4)
\][/tex]
[tex]\[
72 \cdot \left(\frac{4(6x + 4)}{8}\right) = 9 \cdot 4(6x + 4) = 36(6x + 4)
\][/tex]
Thus, the equation becomes:
[tex]\[
24(5x + 4) + 36(6x + 4) = 72 \cdot 22
\][/tex]
3. Distribute within the parentheses:
Distribute the constants through the parentheses:
[tex]\[
24(5x + 4) = 120x + 96
\][/tex]
[tex]\[
36(6x + 4) = 216x + 144
\][/tex]
So, the equation is now:
[tex]\[
120x + 96 + 216x + 144 = 1584
\][/tex]
4. Combine like terms:
Combine the \(x\) terms and the constant terms:
[tex]\[
120x + 216x + 96 + 144 = 1584
\][/tex]
[tex]\[
336x + 240 = 1584
\][/tex]
5. Isolate the \(x\) term:
Move the constant term to the other side of the equation:
[tex]\[
336x = 1584 - 240
\][/tex]
[tex]\[
336x = 1344
\][/tex]
6. Solve for \(x\):
Divide both sides of the equation by 336:
[tex]\[
x = \frac{1344}{336}
\][/tex]
Thus,
[tex]\[
x = 4
\][/tex]
So, the solution to the equation [tex]\(\frac{3(5x + 4)}{9} + \frac{4(6x + 4)}{8} = 22\)[/tex] is [tex]\(x = 4\)[/tex].
