Certainly! Let's solve the equation step-by-step.
### Given Equation
[tex]\[ \frac{32x + 1}{3x} = \frac{82}{9} \][/tex]
### Step 1: Eliminate the fractions
First, we need to eliminate the fractions by cross-multiplying both sides of the equation. This means we will multiply both sides by the denominators on both sides:
[tex]\[ 9 (32x + 1) = 82 (3x) \][/tex]
### Step 2: Distribute and Simplify
Next, distribute the 9 and the 82 on both sides of the equation:
[tex]\[ 9 \cdot 32x + 9 \cdot 1 = 82 \cdot 3x \][/tex]
This simplifies to:
[tex]\[ 288x + 9 = 246x \][/tex]
### Step 3: Isolate the variable term
Now, we need to get all the terms containing [tex]\( x \)[/tex] on one side of the equation. Subtract [tex]\( 246x \)[/tex] from both sides:
[tex]\[ 288x + 9 - 246x = 246x - 246x \][/tex]
This simplifies to:
[tex]\[ 42x + 9 = 0 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting 9 from both sides:
[tex]\[ 42x = -9 \][/tex]
Finally, divide both sides by 42:
[tex]\[ x = \frac{-9}{42} \][/tex]
Simplify the fraction if possible:
[tex]\[ x = -\frac{9}{42} = -\frac{3}{14} \][/tex]
By converting [tex]\(-\frac{3}{14}\)[/tex] to a decimal, we get:
[tex]\[ x \approx -0.214285714285714 \][/tex]
### Final Answer
[tex]\[ x = -0.214285714285714 \][/tex]
This is the solution to the given equation.
