Solve 20(1/2x+1/4)-(-x-1/2)-(-1/3x-frac{3}{2})=frac{5}{6}

$$20 ( \frac { 1 } { 2 } x + \frac { 1 } { 4 } ) - ( - x - \frac { 1 } { 2 } ) - ( - \frac { 1 } { 3 } x - \frac { 3 } { 2 } ) = \frac { 5 } { 6 }$$

Solve for x

$$x=-\frac{37}{68}\approx -0.544117647$$

Steps for Solving Linear Equation

Use the distributive property to multiply 20 by \frac{1}{2}x+\frac{1}{4}.

$$20\times \frac{1}{2}x+20\times \frac{1}{4}-\left(-x-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Multiply 20 and \frac{1}{2} to get \frac{20}{2}.

$$\frac{20}{2}x+20\times \frac{1}{4}-\left(-x-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Divide 20 by 2 to get 10.

$$10x+20\times \frac{1}{4}-\left(-x-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Multiply 20 and \frac{1}{4} to get \frac{20}{4}.

$$10x+\frac{20}{4}-\left(-x-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Divide 20 by 4 to get 5.

$$10x+5-\left(-x-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

To find the opposite of -x-\frac{1}{2}, find the opposite of each term.

$$10x+5-\left(-x\right)-\left(-\frac{1}{2}\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$10x+5-\left(-x\right)+\frac{1}{2}-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Convert 5 to fraction \frac{10}{2}.

$$10x+\frac{10}{2}-\left(-x\right)+\frac{1}{2}-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Since \frac{10}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$10x+\frac{10+1}{2}-\left(-x\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

Add 10 and 1 to get 11.

$$10x+\frac{11}{2}-\left(-x\right)-\left(-\frac{1}{3}x-\frac{3}{2}\right)=\frac{5}{6}$$

To find the opposite of -\frac{1}{3}x-\frac{3}{2}, find the opposite of each term.

$$10x+\frac{11}{2}-\left(-x\right)-\left(-\frac{1}{3}x\right)-\left(-\frac{3}{2}\right)=\frac{5}{6}$$

The opposite of -\frac{1}{3}x is \frac{1}{3}x.

$$10x+\frac{11}{2}-\left(-x\right)+\frac{1}{3}x-\left(-\frac{3}{2}\right)=\frac{5}{6}$$

The opposite of -\frac{3}{2} is \frac{3}{2}.

$$10x+\frac{11}{2}-\left(-x\right)+\frac{1}{3}x+\frac{3}{2}=\frac{5}{6}$$

Combine 10x and \frac{1}{3}x to get \frac{31}{3}x.

$$\frac{31}{3}x+\frac{11}{2}-\left(-x\right)+\frac{3}{2}=\frac{5}{6}$$

Since \frac{11}{2} and \frac{3}{2} have the same denominator, add them by adding their numerators.

$$\frac{31}{3}x+\frac{11+3}{2}-\left(-x\right)=\frac{5}{6}$$

Add 11 and 3 to get 14.

$$\frac{31}{3}x+\frac{14}{2}-\left(-x\right)=\frac{5}{6}$$

Divide 14 by 2 to get 7.

$$\frac{31}{3}x+7-\left(-x\right)=\frac{5}{6}$$

Subtract 7 from both sides.

$$\frac{31}{3}x-\left(-x\right)=\frac{5}{6}-7$$

Convert 7 to fraction \frac{42}{6}.

$$\frac{31}{3}x-\left(-x\right)=\frac{5}{6}-\frac{42}{6}$$

Since \frac{5}{6} and \frac{42}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{31}{3}x-\left(-x\right)=\frac{5-42}{6}$$

Subtract 42 from 5 to get -37.

$$\frac{31}{3}x-\left(-x\right)=-\frac{37}{6}$$

Multiply -1 and -1 to get 1.

$$\frac{31}{3}x+x=-\frac{37}{6}$$

Combine \frac{31}{3}x and x to get \frac{34}{3}x.

$$\frac{34}{3}x=-\frac{37}{6}$$

Multiply both sides by \frac{3}{34}, the reciprocal of \frac{34}{3}.

$$x=-\frac{37}{6}\times \frac{3}{34}$$

Multiply -\frac{37}{6} times \frac{3}{34} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-37\times 3}{6\times 34}$$

Do the multiplications in the fraction \frac{-37\times 3}{6\times 34}.

$$x=\frac{-111}{204}$$

Reduce the fraction \frac{-111}{204} to lowest terms by extracting and canceling out 3.

$$x=-\frac{37}{68}$$