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1. \( \frac{3 x}{2}-\frac{x}{4}=1 \frac{1}{4} \) 3. \( \frac{3}{5}-\frac{7}{x}=\frac{7}{4}-\frac{5}{4 x} \) 5. \( \frac{1-x}{2}+\frac{2-x}{3}+1 \frac{1}{3}=0 \) 7. \( \frac{x-1}{2}-\frac{1}{2}=\frac{2 x-5}{2} \)

Ask by Mitchell Parsons. in South Africa
Feb 18,2025

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Answer

The solutions to the equations are: 1. \( x = 1 \) 2. \( x = -5 \) 3. \( x = 3 \) 4. \( x = 3 \)

Solution

Solve the equation \( \frac{3x}{2}-\frac{x}{4}=1\frac{1}{4} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{3x}{2}-\frac{x}{4}=1\frac{1}{4}\) - step1: Convert the expressions: \(\frac{3x}{2}-\frac{x}{4}=\frac{5}{4}\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{3x}{2}-\frac{x}{4}\right)\times 4=\frac{5}{4}\times 4\) - step3: Simplify the equation: \(5x=5\) - step4: Divide both sides: \(\frac{5x}{5}=\frac{5}{5}\) - step5: Divide the numbers: \(x=1\) Solve the equation \( \frac{3}{5}-\frac{7}{x}=\frac{7}{4}-\frac{5}{4x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{3}{5}-\frac{7}{x}=\frac{7}{4}-\frac{5}{4x}\) - step1: Find the domain: \(\frac{3}{5}-\frac{7}{x}=\frac{7}{4}-\frac{5}{4x},x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{3}{5}-\frac{7}{x}\right)\times 20x=\left(\frac{7}{4}-\frac{5}{4x}\right)\times 20x\) - step3: Simplify the equation: \(12x-140=35x-25\) - step4: Move the expression to the left side: \(12x-35x=-25+140\) - step5: Add and subtract: \(-23x=-25+140\) - step6: Add and subtract: \(-23x=115\) - step7: Change the signs: \(23x=-115\) - step8: Divide both sides: \(\frac{23x}{23}=\frac{-115}{23}\) - step9: Divide the numbers: \(x=-5\) - step10: Check if the solution is in the defined range: \(x=-5,x\neq 0\) - step11: Find the intersection: \(x=-5\) Solve the equation \( \frac{1-x}{2}+\frac{2-x}{3}+1\frac{1}{3}=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{1-x}{2}+\frac{2-x}{3}+1\frac{1}{3}=0\) - step1: Convert the expressions: \(\frac{1-x}{2}+\frac{2-x}{3}+\frac{4}{3}=0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{1-x}{2}+\frac{2-x}{3}+\frac{4}{3}\right)\times 6=0\times 6\) - step3: Simplify the equation: \(15-5x=0\) - step4: Move the constant to the right side: \(-5x=0-15\) - step5: Remove 0: \(-5x=-15\) - step6: Change the signs: \(5x=15\) - step7: Divide both sides: \(\frac{5x}{5}=\frac{15}{5}\) - step8: Divide the numbers: \(x=3\) Solve the equation \( \frac{x-1}{2}-\frac{1}{2}=\frac{2x-5}{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x-1}{2}-\frac{1}{2}=\frac{2x-5}{2}\) - step1: Multiply both sides of the equation by LCD: \(\left(\frac{x-1}{2}-\frac{1}{2}\right)\times 2=\frac{2x-5}{2}\times 2\) - step2: Simplify the equation: \(x-2=2x-5\) - step3: Move the expression to the left side: \(x-2x=-5+2\) - step4: Add and subtract: \(-x=-5+2\) - step5: Add and subtract: \(-x=-3\) - step6: Change the signs: \(x=3\) The solutions to the given equations are: 1. \( x = 1 \) 2. \( x = -5 \) 3. \( x = 3 \) 4. \( x = 3 \)

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To solve the first equation, \( \frac{3x}{2} - \frac{x}{4} = 1 \frac{1}{4} \), let's convert the mixed number into an improper fraction: \( 1 \frac{1}{4} = \frac{5}{4} \). Now, multiply every term by 4 to eliminate the fractions: \[ 4 \left( \frac{3x}{2} \right) - 4 \left( \frac{x}{4} \right) = 4 \cdot \frac{5}{4} \] This simplifies to: \[ 6x - x = 5 \] So, \( 5x = 5 \) or \( x = 1 \). For the third equation, \( \frac{3}{5} - \frac{7}{x} = \frac{7}{4} - \frac{5}{4x} \), we first find a common denominator for the fractions. Multiplying through by \( 20x \) (the least common multiple of the denominators) will do the trick. This yields: \[ 12x - 140 = 35x - 25 \] Rearranging gives \( 25 = 23x \), or \( x = \frac{25}{23} \). For the fifth equation, we need to deal with \( \frac{1 - x}{2} + \frac{2 - x}{3} + 1 \frac{1}{3} = 0 \). Transform \( 1 \frac{1}{3} \) into an improper fraction: \( \frac{4}{3} \). We clear the fractions by multiplying by 6 (LCM of 2 and 3): \[ 3(1 - x) + 2(2 - x) + 8 = 0 \] This simplifies to \( 3 - 3x + 4 - 2x + 8 = 0 \), resulting in \( 15 - 5x = 0 \), or \( x = 3 \). For the seventh equation, \( \frac{x - 1}{2} - \frac{1}{2} = \frac{2x - 5}{2} \), we can multiply through by 2 to eliminate the fractions: \[ x - 1 - 1 = 2x - 5 \] This simplifies to \( x - 2 = 2x - 5 \). Rearranging gives \( x = 3 \). To sum it up, \( x \) values are: 1: \( x = 1 \) 3: \( x = \frac{25}{23} \) 5: \( x = 3 \) 7: \( x = 3 \)

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