Solve (3/4-5/2x)^-1=1/3

$$( \frac { 3 } { 4 } - \frac { 5 } { 2 x } ) ^ { - 1 } = \frac { 1 } { 3 }$$

Solve for x

$$x = -\frac{10}{9} = -1\frac{1}{9} \approx -1.111111111$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2x is 4x. Multiply \frac{3}{4} times \frac{x}{x}. Multiply \frac{5}{2x} times \frac{2}{2}.

$$\left(\frac{3x}{4x}-\frac{5\times 2}{4x}\right)^{-1}=\frac{1}{3}$$

Since \frac{3x}{4x} and \frac{5\times 2}{4x} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3x-5\times 2}{4x}\right)^{-1}=\frac{1}{3}$$

Do the multiplications in 3x-5\times 2.

$$\left(\frac{3x-10}{4x}\right)^{-1}=\frac{1}{3}$$

To raise \frac{3x-10}{4x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3x-10\right)^{-1}}{\left(4x\right)^{-1}}=\frac{1}{3}$$

Expand \left(4x\right)^{-1}.

$$\frac{\left(3x-10\right)^{-1}}{4^{-1}x^{-1}}=\frac{1}{3}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{\left(3x-10\right)^{-1}}{\frac{1}{4}x^{-1}}=\frac{1}{3}$$

Reorder the terms.

$$\frac{1}{\frac{1}{4}\times \frac{1}{x}\left(3x-10\right)}=\frac{1}{3}$$

Multiply \frac{1}{4} times \frac{1}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{\frac{1}{4x}\left(3x-10\right)}=\frac{1}{3}$$

Express \frac{1}{4x}\left(3x-10\right) as a single fraction.

$$\frac{1}{\frac{3x-10}{4x}}=\frac{1}{3}$$

Variable x cannot be equal to 0 since division by zero is not defined. Divide 1 by \frac{3x-10}{4x} by multiplying 1 by the reciprocal of \frac{3x-10}{4x}.

$$\frac{4x}{3x-10}=\frac{1}{3}$$

Variable x cannot be equal to \frac{10}{3} since division by zero is not defined. Multiply both sides of the equation by 3\left(3x-10\right), the least common multiple of 3x-10,3.

$$3\times 4x=3x-10$$

Multiply 3 and 4 to get 12.

$$12x=3x-10$$

Subtract 3x from both sides.

$$12x-3x=-10$$

Combine 12x and -3x to get 9x.

$$9x=-10$$

Divide both sides by 9.

$$x=\frac{-10}{9}$$

Fraction \frac{-10}{9} can be rewritten as -\frac{10}{9} by extracting the negative sign.

$$x=-\frac{10}{9}$$