Question
Which of the following represents the solution of \( \frac{3}{2}=\frac{3}{2 x}-\frac{6}{5 x} ? \)
Ask by Montgomery Mccarthy. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( x = \frac{1}{5} \) or \( x = 0.2 \).
Solution
Solve the equation \( \frac{3}{2}=\frac{3}{2x}-\frac{6}{5x} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}\)
- step1: Find the domain:
\(\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x},x\neq 0\)
- step2: Swap the sides:
\(\frac{3}{2x}-\frac{6}{5x}=\frac{3}{2}\)
- step3: Multiply both sides of the equation by LCD:
\(\left(\frac{3}{2x}-\frac{6}{5x}\right)\times 10x=\frac{3}{2}\times 10x\)
- step4: Simplify the equation:
\(3=15x\)
- step5: Swap the sides:
\(15x=3\)
- step6: Divide both sides:
\(\frac{15x}{15}=\frac{3}{15}\)
- step7: Divide the numbers:
\(x=\frac{1}{5}\)
- step8: Check if the solution is in the defined range:
\(x=\frac{1}{5},x\neq 0\)
- step9: Find the intersection:
\(x=\frac{1}{5}\)
The solution to the equation \( \frac{3}{2}=\frac{3}{2x}-\frac{6}{5x} \) is \( x = \frac{1}{5} \) or \( x = 0.2 \).
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Simplify this solution Bonus Knowledge
To solve the equation \( \frac{3}{2}=\frac{3}{2 x}-\frac{6}{5 x} \), first, we want to eliminate the fractions. Start by finding a common denominator for the right side, which is \( 10x \). The equation becomes: \[ \frac{3 \cdot 5}{10} = \frac{3 \cdot 5 - 6 \cdot 2}{10x} \] This simplifies to: \[ \frac{15}{10} = \frac{15 - 12}{10x} \] So, rewriting it gives us: \[ \frac{15}{10} = \frac{3}{10x} \] Cross-multiplying gives: \[ 15 \cdot 10x = 3 \cdot 10 \] Which simplifies to: \[ 150x = 30 \] Dividing both sides by 150 results in: \[ x = \frac{30}{150} = \frac{1}{5} \] So, the solution is \( x = \frac{1}{5} \).
