Solving the Fraction Equation for x

The problem asks us to find the value of the variable 'x' that satisfies the given algebraic equation involving fractions. The equation is:

$$ \frac{4}{x} - \frac{3}{2x} = 20 $$

To solve for 'x', we need to combine the terms on the left side of the equation and then isolate 'x'.

Step-by-Step Solution

1. Find a Common Denominator: The terms on the left side have denominators '$x$' and '$2x$'. The least common denominator (LCD) for these two terms is '$2x$'.
2. Rewrite Fractions: Convert the first fraction, $\frac{4}{x}$, so it has the denominator '$2x$'. We multiply the numerator and denominator by 2: $$ \frac{4}{x} = \frac{4 \times 2}{x \times 2} = \frac{8}{2x} $$
3. Substitute and Combine: Now substitute this back into the original equation: $$ \frac{8}{2x} - \frac{3}{2x} = 20 $$ Since the denominators are the same, we can combine the numerators: $$ \frac{8 - 3}{2x} = 20 $$ $$ \frac{5}{2x} = 20 $$
4. Isolate the Variable Term: To isolate the term containing 'x', we can multiply both sides of the equation by '$2x$' (assuming $x \neq 0$): $$ 5 = 20 \times (2x) $$ $$ 5 = 40x $$
5. Solve for x: Now, divide both sides by 40 to find the value of 'x': $$ x = \frac{5}{40} $$
6. Simplify the Result: Reduce the fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor, which is 5: $$ x = \frac{5 \div 5}{40 \div 5} = \frac{1}{8} $$

Final Answer Verification

We found that $x = \frac{1}{8}$. Let's check this by plugging it back into the original equation:

$$ \frac{4}{(\frac{1}{8})} - \frac{3}{2(\frac{1}{8})} $$

$$ = (4 \times 8) - \frac{3}{(\frac{2}{8})} $$

$$ = 32 - \frac{3}{(\frac{1}{4})} $$

$$ = 32 - (3 \times 4) $$

$$ = 32 - 12 $$

$$ = 20 $$

The result matches the right side of the original equation, confirming our solution is correct.

Therefore, the value of x is $\frac{1}{8}$.