To solve the given equation:
[tex]\[
\frac{\frac{x}{4} - \frac{3}{5}}{\frac{4}{3} - 7x} = -\frac{3}{20}
\][/tex]
we will follow a step-by-step approach.
### Step 1: Simplify the Numerator and Denominator
First, we break down and simplify the numerator and the denominator of the left-hand side of the equation:
1. Numerator:
[tex]\[
\frac{x}{4} - \frac{3}{5}
\][/tex]
To combine these fractions, we need a common denominator. The common denominator of 4 and 5 is 20:
[tex]\[
\frac{x}{4} = \frac{5x}{20}, \quad -\frac{3}{5} = -\frac{12}{20}
\][/tex]
Thus, the numerator becomes:
[tex]\[
\frac{5x - 12}{20}
\][/tex]
2. Denominator:
[tex]\[
\frac{4}{3} - 7x
\][/tex]
This expression will remain as it is since it's already simplified.
### Step 2: Substitute Back into the Equation
The equation now simplifies to:
[tex]\[
\frac{\frac{5x - 12}{20}}{\frac{4}{3} - 7x} = -\frac{3}{20}
\][/tex]
### Step 3: Clear the Fraction
To eliminate the fraction on the left side, we'll multiply both sides of the equation by the denominator of the fraction [tex]\(\left(\frac{4}{3} - 7x\right)\)[/tex]:
[tex]\[
\frac{5x - 12}{20} = \left(-\frac{3}{20}\right) \left(\frac{4}{3} - 7x\right)
\][/tex]
### Step 4: Simplify Further
Next, we'll distribute [tex]\(-\frac{3}{20}\)[/tex] on the right-hand side:
[tex]\[
\frac{5x - 12}{20} = -\frac{3}{20} \cdot \frac{4}{3} + \frac{3}{20} \cdot 7x
\][/tex]
Simplify the right-hand side:
[tex]\[
\frac{5x - 12}{20} = -\frac{3 \cdot 4}{20 \cdot 3} + \frac{3 \cdot 7x}{20}
\][/tex]
[tex]\[
\frac{5x - 12}{20} = -\frac{12}{60} + \frac{21x}{20}
\][/tex]
[tex]\[
\frac{5x - 12}{20} = -\frac{1}{5} + \frac{21x}{20}
\][/tex]
[tex]\[
\frac{5x - 12}{20} = \frac{21x - 4}{20}
\][/tex]
### Step 5: Equate the Numerators
Since the denominators are the same, we can now equate the numerators:
[tex]\[
5x - 12 = 21x - 4
\][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], we move all [tex]\(x\)[/tex]-terms to one side and constants to the other:
[tex]\[
5x - 21x = -4 + 12
\][/tex]
[tex]\[
-16x = 8
\][/tex]
[tex]\[
x = \frac{8}{-16}
\][/tex]
[tex]\[
x = -\frac{1}{2}
\][/tex]
So the solution to the equation is:
[tex]\[
x = -0.5
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the given equation is:
[tex]\[
x = -0.5
\][/tex]
