Question
Add. \[ \frac{3}{2 x}+\frac{2}{x-4} \] Simplify your answer as much as possible.
Ask by Hodges Chan. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified sum is \( \frac{7x - 12}{2x^{2} - 8x} \).
Solution
Calculate or simplify the expression \( \frac{3}{2x}+\frac{2}{x-4} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{3}{2x}+\frac{2}{x-4}\)
- step1: Reduce fractions to a common denominator:
\(\frac{3\left(x-4\right)}{2x\left(x-4\right)}+\frac{2\times 2x}{\left(x-4\right)\times 2x}\)
- step2: Reorder the terms:
\(\frac{3\left(x-4\right)}{2x\left(x-4\right)}+\frac{2\times 2x}{2\left(x-4\right)x}\)
- step3: Rewrite the expression:
\(\frac{3\left(x-4\right)}{2x\left(x-4\right)}+\frac{2\times 2x}{2x\left(x-4\right)}\)
- step4: Transform the expression:
\(\frac{3\left(x-4\right)+2\times 2x}{2x\left(x-4\right)}\)
- step5: Multiply the terms:
\(\frac{3x-12+2\times 2x}{2x\left(x-4\right)}\)
- step6: Multiply the terms:
\(\frac{3x-12+4x}{2x\left(x-4\right)}\)
- step7: Add the terms:
\(\frac{7x-12}{2x\left(x-4\right)}\)
- step8: Multiply the terms:
\(\frac{7x-12}{2x^{2}-8x}\)
The sum of the given fractions is \(\frac{7x-12}{2x^{2}-8x}\).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Mind Expander
To add the fractions \(\frac{3}{2x}\) and \(\frac{2}{x-4}\), we first find a common denominator. The least common denominator (LCD) for \(2x\) and \(x-4\) is \(2x(x-4)\). Now, we rewrite each fraction with the common denominator: \[ \frac{3}{2x} = \frac{3(x-4)}{2x(x-4)} = \frac{3x - 12}{2x(x-4)} \] \[ \frac{2}{x-4} = \frac{2 \cdot 2x}{(x-4) \cdot 2x} = \frac{4x}{2x(x-4)} \] Now we can add the two fractions: \[ \frac{3x - 12 + 4x}{2x(x-4)} = \frac{7x - 12}{2x(x-4)} \] So the simplified result is: \[ \frac{7x - 12}{2x(x-4)} \]
