Solve 1/3(1/5x-3)=1/2(2-1/x)

$$\frac { 1 } { 3 } ( \frac { 1 } { 5 x } - 3 ) = \frac { 1 } { 2 } ( 2 - \frac { 1 } { x } )$$

Solve for x

$$x=\frac{17}{60}\approx 0.283333333$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 30x, the least common multiple of 3,5x,2,x.

$$\frac{1}{3}\left(\frac{1}{5x}-3\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{5x}{5x}.

$$\frac{1}{3}\left(\frac{1}{5x}-\frac{3\times 5x}{5x}\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Since \frac{1}{5x} and \frac{3\times 5x}{5x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{1-3\times 5x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Do the multiplications in 1-3\times 5x.

$$\frac{1}{3}\times \frac{1-15x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Multiply \frac{1}{3} and 30 to get 10.

$$10\times \frac{1-15x}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express 10\times \frac{1-15x}{5x} as a single fraction.

$$\frac{10\left(1-15x\right)}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\left(-15x+1\right)}{x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express \frac{2\left(-15x+1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(\frac{2x}{x}-\frac{1}{x}\right)\times 30x$$

Since \frac{2x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\times \frac{2x-1}{x}\times 30x$$

Multiply \frac{1}{2} and 30 to get 15.

$$\frac{2\left(-15x+1\right)x}{x}=15\times \frac{2x-1}{x}x$$

Express 15\times \frac{2x-1}{x} as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)}{x}x$$

Express \frac{15\left(2x-1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 2 by -15x+1.

$$\frac{\left(-30x+2\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply -30x+2 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 15 by 2x-1.

$$\frac{-30x^{2}+2x}{x}=\frac{\left(30x-15\right)x}{x}$$

Use the distributive property to multiply 30x-15 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{30x^{2}-15x}{x}$$

Subtract \frac{30x^{2}-15x}{x} from both sides.

$$\frac{-30x^{2}+2x}{x}-\frac{30x^{2}-15x}{x}=0$$

Since \frac{-30x^{2}+2x}{x} and \frac{30x^{2}-15x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-30x^{2}+2x-\left(30x^{2}-15x\right)}{x}=0$$

Do the multiplications in -30x^{2}+2x-\left(30x^{2}-15x\right).

$$\frac{-30x^{2}+2x-30x^{2}+15x}{x}=0$$

Combine like terms in -30x^{2}+2x-30x^{2}+15x.

$$\frac{-60x^{2}+17x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-60x^{2}+17x=0$$

Factor out x.

$$x\left(-60x+17\right)=0$$

To find equation solutions, solve x=0 and -60x+17=0.

$$x=0 x=\frac{17}{60}$$

Variable x cannot be equal to 0.

$$x=\frac{17}{60}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 30x, the least common multiple of 3,5x,2,x.

$$\frac{1}{3}\left(\frac{1}{5x}-3\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{5x}{5x}.

$$\frac{1}{3}\left(\frac{1}{5x}-\frac{3\times 5x}{5x}\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Since \frac{1}{5x} and \frac{3\times 5x}{5x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{1-3\times 5x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Do the multiplications in 1-3\times 5x.

$$\frac{1}{3}\times \frac{1-15x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Multiply \frac{1}{3} and 30 to get 10.

$$10\times \frac{1-15x}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express 10\times \frac{1-15x}{5x} as a single fraction.

$$\frac{10\left(1-15x\right)}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\left(-15x+1\right)}{x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express \frac{2\left(-15x+1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(\frac{2x}{x}-\frac{1}{x}\right)\times 30x$$

Since \frac{2x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\times \frac{2x-1}{x}\times 30x$$

Multiply \frac{1}{2} and 30 to get 15.

$$\frac{2\left(-15x+1\right)x}{x}=15\times \frac{2x-1}{x}x$$

Express 15\times \frac{2x-1}{x} as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)}{x}x$$

Express \frac{15\left(2x-1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 2 by -15x+1.

$$\frac{\left(-30x+2\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply -30x+2 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 15 by 2x-1.

$$\frac{-30x^{2}+2x}{x}=\frac{\left(30x-15\right)x}{x}$$

Use the distributive property to multiply 30x-15 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{30x^{2}-15x}{x}$$

Subtract \frac{30x^{2}-15x}{x} from both sides.

$$\frac{-30x^{2}+2x}{x}-\frac{30x^{2}-15x}{x}=0$$

Since \frac{-30x^{2}+2x}{x} and \frac{30x^{2}-15x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-30x^{2}+2x-\left(30x^{2}-15x\right)}{x}=0$$

Do the multiplications in -30x^{2}+2x-\left(30x^{2}-15x\right).

$$\frac{-30x^{2}+2x-30x^{2}+15x}{x}=0$$

Combine like terms in -30x^{2}+2x-30x^{2}+15x.

$$\frac{-60x^{2}+17x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-60x^{2}+17x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -60 for a, 17 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}}}{2\left(-60\right)}$$

Take the square root of 17^{2}.

$$x=\frac{-17±17}{2\left(-60\right)}$$

Multiply 2 times -60.

$$x=\frac{-17±17}{-120}$$

Now solve the equation x=\frac{-17±17}{-120} when ± is plus. Add -17 to 17.

$$x=\frac{0}{-120}$$

Divide 0 by -120.

$$x=0$$

Now solve the equation x=\frac{-17±17}{-120} when ± is minus. Subtract 17 from -17.

$$x=-\frac{34}{-120}$$

Reduce the fraction \frac{-34}{-120} to lowest terms by extracting and canceling out 2.

$$x=\frac{17}{60}$$

The equation is now solved.

$$x=0 x=\frac{17}{60}$$

Variable x cannot be equal to 0.

$$x=\frac{17}{60}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 30x, the least common multiple of 3,5x,2,x.

$$\frac{1}{3}\left(\frac{1}{5x}-3\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{5x}{5x}.

$$\frac{1}{3}\left(\frac{1}{5x}-\frac{3\times 5x}{5x}\right)\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Since \frac{1}{5x} and \frac{3\times 5x}{5x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{1-3\times 5x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Do the multiplications in 1-3\times 5x.

$$\frac{1}{3}\times \frac{1-15x}{5x}\times 30x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Multiply \frac{1}{3} and 30 to get 10.

$$10\times \frac{1-15x}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express 10\times \frac{1-15x}{5x} as a single fraction.

$$\frac{10\left(1-15x\right)}{5x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\left(-15x+1\right)}{x}x=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

Express \frac{2\left(-15x+1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(2-\frac{1}{x}\right)\times 30x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\left(\frac{2x}{x}-\frac{1}{x}\right)\times 30x$$

Since \frac{2x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{1}{2}\times \frac{2x-1}{x}\times 30x$$

Multiply \frac{1}{2} and 30 to get 15.

$$\frac{2\left(-15x+1\right)x}{x}=15\times \frac{2x-1}{x}x$$

Express 15\times \frac{2x-1}{x} as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)}{x}x$$

Express \frac{15\left(2x-1\right)}{x}x as a single fraction.

$$\frac{2\left(-15x+1\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 2 by -15x+1.

$$\frac{\left(-30x+2\right)x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply -30x+2 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{15\left(2x-1\right)x}{x}$$

Use the distributive property to multiply 15 by 2x-1.

$$\frac{-30x^{2}+2x}{x}=\frac{\left(30x-15\right)x}{x}$$

Use the distributive property to multiply 30x-15 by x.

$$\frac{-30x^{2}+2x}{x}=\frac{30x^{2}-15x}{x}$$

Subtract \frac{30x^{2}-15x}{x} from both sides.

$$\frac{-30x^{2}+2x}{x}-\frac{30x^{2}-15x}{x}=0$$

Since \frac{-30x^{2}+2x}{x} and \frac{30x^{2}-15x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-30x^{2}+2x-\left(30x^{2}-15x\right)}{x}=0$$

Do the multiplications in -30x^{2}+2x-\left(30x^{2}-15x\right).

$$\frac{-30x^{2}+2x-30x^{2}+15x}{x}=0$$

Combine like terms in -30x^{2}+2x-30x^{2}+15x.

$$\frac{-60x^{2}+17x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-60x^{2}+17x=0$$

Divide both sides by -60.

$$\frac{-60x^{2}+17x}{-60}=\frac{0}{-60}$$

Dividing by -60 undoes the multiplication by -60.

$$x^{2}+\frac{17}{-60}x=\frac{0}{-60}$$

Divide 17 by -60.

$$x^{2}-\frac{17}{60}x=\frac{0}{-60}$$

Divide 0 by -60.

$$x^{2}-\frac{17}{60}x=0$$

Divide -\frac{17}{60}, the coefficient of the x term, by 2 to get -\frac{17}{120}. Then add the square of -\frac{17}{120} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{60}x+\left(-\frac{17}{120}\right)^{2}=\left(-\frac{17}{120}\right)^{2}$$

Square -\frac{17}{120} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{60}x+\frac{289}{14400}=\frac{289}{14400}$$

Factor x^{2}-\frac{17}{60}x+\frac{289}{14400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{120}\right)^{2}=\frac{289}{14400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{120}\right)^{2}}=\sqrt{\frac{289}{14400}}$$

Simplify.

$$x-\frac{17}{120}=\frac{17}{120} x-\frac{17}{120}=-\frac{17}{120}$$

Add \frac{17}{120} to both sides of the equation.

$$x=\frac{17}{60} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{17}{60}$$