Question

question
solve for x.
\\(\dfrac{x - 8}{x} + \dfrac{1}{2x} = \dfrac{-1}{2}\\)
answer
attempt 1 out of 2
x =

Explanation:

Step1: Find common denominator

The denominators are \(x\), \(2x\), and \(2\). The least common denominator (LCD) is \(2x\). Multiply each term by \(2x\) to eliminate fractions:

$$ 2x \cdot \frac{x - 8}{x} + 2x \cdot \frac{1}{2x} = 2x \cdot \frac{-1}{2} $$

Step2: Simplify each term

Simplify each term after multiplying by \(2x\):

• For the first term: \(2x \cdot \frac{x - 8}{x} = 2(x - 8) = 2x - 16\)
• For the second term: \(2x \cdot \frac{1}{2x} = 1\)
• For the third term: \(2x \cdot \frac{-1}{2} = -x\)

So the equation becomes:

$$ 2x - 16 + 1 = -x $$

Step3: Combine like terms

Combine the constant terms on the left side:

$$ 2x - 15 = -x $$

Step4: Solve for \(x\)

Add \(x\) to both sides:

$$ 2x + x - 15 = -x + x \implies 3x - 15 = 0 $$

Add \(15\) to both sides:

$$ 3x - 15 + 15 = 0 + 15 \implies 3x = 15 $$

Divide both sides by \(3\):

$$ x = \frac{15}{3} = 5 $$

Step5: Check for extraneous solutions

Since the original equation has a denominator of \(x\) and \(2x\), \(x
eq 0\). Substitute \(x = 5\) back into the original equation to check:

• Left side: \(\frac{5 - 8}{5} + \frac{1}{2 \cdot 5} = \frac{-3}{5} + \frac{1}{10} = \frac{-6}{10} + \frac{1}{10} = \frac{-5}{10} = -\frac{1}{2}\)
• Right side: \(\frac{-1}{2}\)

Both sides are equal, so \(x = 5\) is a valid solution.

Answer:

\(x = 5\)