Solve (1/x+1/12)^-1+1=4

$$( \frac { 1 } { x } + \frac { 1 } { 12 } ) ^ { - 1 } + 1 = 4$$

Solve for x

$$x=4$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 12 is 12x. Multiply \frac{1}{x} times \frac{12}{12}. Multiply \frac{1}{12} times \frac{x}{x}.

$$\left(\frac{12}{12x}+\frac{x}{12x}\right)^{-1}+1=4$$

Since \frac{12}{12x} and \frac{x}{12x} have the same denominator, add them by adding their numerators.

$$\left(\frac{12+x}{12x}\right)^{-1}+1=4$$

To raise \frac{12+x}{12x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(12+x\right)^{-1}}{\left(12x\right)^{-1}}+1=4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(12x\right)^{-1}}{\left(12x\right)^{-1}}.

$$\frac{\left(12+x\right)^{-1}}{\left(12x\right)^{-1}}+\frac{\left(12x\right)^{-1}}{\left(12x\right)^{-1}}=4$$

Since \frac{\left(12+x\right)^{-1}}{\left(12x\right)^{-1}} and \frac{\left(12x\right)^{-1}}{\left(12x\right)^{-1}} have the same denominator, add them by adding their numerators.

$$\frac{\left(12+x\right)^{-1}+\left(12x\right)^{-1}}{\left(12x\right)^{-1}}=4$$

Expand \left(12x\right)^{-1}.

$$\frac{\left(12+x\right)^{-1}+12^{-1}x^{-1}}{\left(12x\right)^{-1}}=4$$

Calculate 12 to the power of -1 and get \frac{1}{12}.

$$\frac{\left(12+x\right)^{-1}+\frac{1}{12}x^{-1}}{\left(12x\right)^{-1}}=4$$

Expand \left(12x\right)^{-1}.

$$\frac{\left(12+x\right)^{-1}+\frac{1}{12}x^{-1}}{12^{-1}x^{-1}}=4$$

Calculate 12 to the power of -1 and get \frac{1}{12}.

$$\frac{\left(12+x\right)^{-1}+\frac{1}{12}x^{-1}}{\frac{1}{12}x^{-1}}=4$$

Factor the expressions that are not already factored in \frac{\left(12+x\right)^{-1}+\frac{1}{12}x^{-1}}{\frac{1}{12}x^{-1}}.

$$\frac{\frac{1}{12}\left(12\times \frac{1}{x+12}+\frac{1}{x}\right)}{\frac{1}{12}\times \frac{1}{x}}=4$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{12\times \frac{1}{x+12}+\frac{1}{x}}{\left(\frac{1}{12}\right)^{0}\times \frac{1}{x}}=4$$

Express 12\times \frac{1}{x+12} as a single fraction.

$$\frac{\frac{12}{x+12}+\frac{1}{x}}{\left(\frac{1}{12}\right)^{0}\times \frac{1}{x}}=4$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+12 and x is x\left(x+12\right). Multiply \frac{12}{x+12} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+12}{x+12}.

$$\frac{\frac{12x}{x\left(x+12\right)}+\frac{x+12}{x\left(x+12\right)}}{\left(\frac{1}{12}\right)^{0}\times \frac{1}{x}}=4$$

Since \frac{12x}{x\left(x+12\right)} and \frac{x+12}{x\left(x+12\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{12x+x+12}{x\left(x+12\right)}}{\left(\frac{1}{12}\right)^{0}\times \frac{1}{x}}=4$$

Combine like terms in 12x+x+12.

$$\frac{\frac{13x+12}{x\left(x+12\right)}}{\left(\frac{1}{12}\right)^{0}\times \frac{1}{x}}=4$$

Calculate \frac{1}{12} to the power of 0 and get 1.

$$\frac{\frac{13x+12}{x\left(x+12\right)}}{1\times \frac{1}{x}}=4$$

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

$$\frac{\frac{13x+12}{x\left(x+12\right)}}{\frac{1}{x}}=4$$

Variable x cannot be equal to 0 since division by zero is not defined. Divide \frac{13x+12}{x\left(x+12\right)} by \frac{1}{x} by multiplying \frac{13x+12}{x\left(x+12\right)} by the reciprocal of \frac{1}{x}.

$$\frac{\left(13x+12\right)x}{x\left(x+12\right)}=4$$

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+12\right).

$$\left(13x+12\right)x=4x\left(x+12\right)$$

Reorder the terms.

$$x\left(13x+12\right)=4x\left(x+12\right)$$

Use the distributive property to multiply x by 13x+12.

$$13x^{2}+12x=4x\left(x+12\right)$$

Use the distributive property to multiply 4x by x+12.

$$13x^{2}+12x=4x^{2}+48x$$

Subtract 4x^{2} from both sides.

$$13x^{2}+12x-4x^{2}=48x$$

Combine 13x^{2} and -4x^{2} to get 9x^{2}.

$$9x^{2}+12x=48x$$

Subtract 48x from both sides.

$$9x^{2}+12x-48x=0$$

Combine 12x and -48x to get -36x.

$$9x^{2}-36x=0$$