Certainly! Let's solve the given equation step-by-step.
### Step 1: Solve the equation [tex]\( 3 - \frac{1}{4x} = \frac{1}{8x} \)[/tex]
First, let's isolate the variable [tex]\( x \)[/tex]. Start by eliminating the fractions, which can be achieved by finding a common denominator. The common denominator here is [tex]\( 8x \)[/tex]:
Multiplying through by [tex]\( 8x \)[/tex] gives:
[tex]\[ 8x \left( 3 - \frac{1}{4x} \right) = 8x \left( \frac{1}{8x} \right) \][/tex]
Simplifying, we get:
[tex]\[ 8x \cdot 3 - 8x \cdot \frac{1}{4x} = 1 \][/tex]
[tex]\[ 24x - 2 = 1 \][/tex]
### Step 2: Solve the Linear Equation
Next, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[ 24x - 2 = 1 \][/tex]
[tex]\[ 24x = 3 \][/tex]
[tex]\[ x = \frac{3}{24} \][/tex]
[tex]\[ x = \frac{1}{8} \][/tex]
### Step 3: Calculate [tex]\( \frac{1}{x} + 2 \)[/tex]
Now that we have [tex]\( x = \frac{1}{8} \)[/tex], we substitute it to find [tex]\( \frac{1}{x} + 2 \)[/tex]:
[tex]\[ \frac{1}{x} + 2 = \frac{1}{\frac{1}{8}} + 2 \][/tex]
Inverting [tex]\( \frac{1}{8} \)[/tex]:
[tex]\[ \frac{1}{\frac{1}{8}} = 8 \][/tex]
So, we have:
[tex]\[ 8 + 2 = 10 \][/tex]
### Step 4: Calculate the Ratio [tex]\( \left( \frac{1}{x} + 2 \right) : 1 \)[/tex]
We need to express [tex]\( \left( \frac{1}{x} + 2 \right) : 1 \)[/tex] as a ratio:
[tex]\[ 10 : 1 \][/tex]
### Conclusion
Thus, the ratio [tex]\( \left( \frac{1}{x} + 2 \right) \)[/tex] to 1 is [tex]\( 10:1 \)[/tex].
