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For the following equation, a. Write the value or values of \[ \frac{2}{9 x}+\frac{1}{4}=\frac{67}{18 x}-\frac{1}{3} \] a. Write the va ue or values of the veriable that make a den

Ask by Savage Hamilton. in the United States
Jan 26,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

\( x = 6 \)

Solution

Solve the equation \( \frac{2}{9x}+\frac{1}{4}=\frac{67}{18x}-\frac{1}{3} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{2}{9x}+\frac{1}{4}=\frac{67}{18x}-\frac{1}{3}\) - step1: Find the domain: \(\frac{2}{9x}+\frac{1}{4}=\frac{67}{18x}-\frac{1}{3},x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{2}{9x}+\frac{1}{4}\right)\times 36x=\left(\frac{67}{18x}-\frac{1}{3}\right)\times 36x\) - step3: Simplify the equation: \(8+9x=134-12x\) - step4: Move the expression to the left side: \(9x+12x=134-8\) - step5: Add and subtract: \(21x=134-8\) - step6: Add and subtract: \(21x=126\) - step7: Divide both sides: \(\frac{21x}{21}=\frac{126}{21}\) - step8: Divide the numbers: \(x=6\) - step9: Check if the solution is in the defined range: \(x=6,x\neq 0\) - step10: Find the intersection: \(x=6\) The solution to the equation is \( x = 6 \).

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Bonus Knowledge

To solve the equation \( \frac{2}{9x} + \frac{1}{4} = \frac{67}{18x} - \frac{1}{3} \), you’ll first want to find a common denominator to combine the fractions effectively. The least common multiple of the denominators (9x, 4, 18x, and 3) is \(36x\). By rewriting each term with this common denominator, you'll be able to consolidate and solve for \(x\) easily. Once you have solved for \(x\), don’t forget to check your answer against the original equation to ensure it doesn’t involve division by zero. This step is crucial, as it helps avoid invalid solutions where the variable would cause the denominators to become zero!

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