Problem 7 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$\frac{x}{8-2 x} \div \frac{2 x}{4-x}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{4} \), for \( x ≠ 4,0 \) because x in the denominator cannot equal 0 or 4

Step 1: Rewrite the division as multiplication

To divide fractions, we multiply the first fraction by the reciprocal of the second. So, the given problem can be rewritten as \( \frac{x}{8-2 x} \times \frac{4-x}{2 x} \)

Step 2: Multiply the Fractions

Here now we multiply the numerators and the denominators: \( \frac{x(4-x)}{(8-2x)(2x)} \)

Step 3: Simplify the Result

After performing the multiplication, the resulting expression is \( \frac{4x-x^2}{16x-4x^2} \). We can simplify further by factoring out \( x \) from the numerator and denominator to obtain \( \frac{x(4-x)}{4x(4-x)} \). Since the same term appears in the numerator and denumerator, we can simplify to get the final answer \( \frac{x}{4x} = \frac{1}{4} \) if and only if \( x ≠ 4,0 \) because x in the denominator cannot equal 0 and 4