Solve 8[1/2x-1/4(x+1)]-1/2=15[1/2-frac{2}{3}(x-frac{1}{2})]+2x

$$8 [ \frac { 1 } { 2 } x - \frac { 1 } { 4 } ( x + 1 ) ] - \frac { 1 } { 2 } = 15 [ \frac { 1 } { 2 } - \frac { 2 } { 3 } ( x - \frac { 1 } { 2 } ) ] + 2 x$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps for Solving Linear Equation

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

$$8\left(\frac{1}{2}x-\frac{1}{4}x-\frac{1}{4}\right)-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Combine \frac{1}{2}x and -\frac{1}{4}x to get \frac{1}{4}x.

$$8\left(\frac{1}{4}x-\frac{1}{4}\right)-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Use the distributive property to multiply 8 by \frac{1}{4}x-\frac{1}{4}.

$$8\times \frac{1}{4}x+8\left(-\frac{1}{4}\right)-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Multiply 8 and \frac{1}{4} to get \frac{8}{4}.

$$\frac{8}{4}x+8\left(-\frac{1}{4}\right)-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Divide 8 by 4 to get 2.

$$2x+8\left(-\frac{1}{4}\right)-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Express 8\left(-\frac{1}{4}\right) as a single fraction.

$$2x+\frac{8\left(-1\right)}{4}-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Multiply 8 and -1 to get -8.

$$2x+\frac{-8}{4}-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Divide -8 by 4 to get -2.

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

Convert -2 to fraction -\frac{4}{2}.

$$2x-\frac{4}{2}-\frac{1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Since -\frac{4}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$2x+\frac{-4-1}{2}=15\left(\frac{1}{2}-\frac{2}{3}\left(x-\frac{1}{2}\right)\right)+2x$$

Subtract 1 from -4 to get -5.

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

Use the distributive property to multiply -\frac{2}{3} by x-\frac{1}{2}.

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

Multiply -\frac{2}{3} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{-2\left(-1\right)}{3\times 2}.

$$2x-\frac{5}{2}=15\left(\frac{1}{2}-\frac{2}{3}x+\frac{2}{6}\right)+2x$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

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

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$2x-\frac{5}{2}=15\left(\frac{3}{6}-\frac{2}{3}x+\frac{2}{6}\right)+2x$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, add them by adding their numerators.

$$2x-\frac{5}{2}=15\left(\frac{3+2}{6}-\frac{2}{3}x\right)+2x$$

Add 3 and 2 to get 5.

$$2x-\frac{5}{2}=15\left(\frac{5}{6}-\frac{2}{3}x\right)+2x$$

Use the distributive property to multiply 15 by \frac{5}{6}-\frac{2}{3}x.

$$2x-\frac{5}{2}=15\times \frac{5}{6}+15\left(-\frac{2}{3}\right)x+2x$$

Express 15\times \frac{5}{6} as a single fraction.

$$2x-\frac{5}{2}=\frac{15\times 5}{6}+15\left(-\frac{2}{3}\right)x+2x$$

Multiply 15 and 5 to get 75.

$$2x-\frac{5}{2}=\frac{75}{6}+15\left(-\frac{2}{3}\right)x+2x$$

Reduce the fraction \frac{75}{6} to lowest terms by extracting and canceling out 3.

$$2x-\frac{5}{2}=\frac{25}{2}+15\left(-\frac{2}{3}\right)x+2x$$

Express 15\left(-\frac{2}{3}\right) as a single fraction.

$$2x-\frac{5}{2}=\frac{25}{2}+\frac{15\left(-2\right)}{3}x+2x$$

Multiply 15 and -2 to get -30.

$$2x-\frac{5}{2}=\frac{25}{2}+\frac{-30}{3}x+2x$$

Divide -30 by 3 to get -10.

$$2x-\frac{5}{2}=\frac{25}{2}-10x+2x$$

Combine -10x and 2x to get -8x.

$$2x-\frac{5}{2}=\frac{25}{2}-8x$$

Add 8x to both sides.

$$2x-\frac{5}{2}+8x=\frac{25}{2}$$

Combine 2x and 8x to get 10x.

$$10x-\frac{5}{2}=\frac{25}{2}$$

Add \frac{5}{2} to both sides.

$$10x=\frac{25}{2}+\frac{5}{2}$$

Since \frac{25}{2} and \frac{5}{2} have the same denominator, add them by adding their numerators.

$$10x=\frac{25+5}{2}$$

Add 25 and 5 to get 30.

$$10x=\frac{30}{2}$$

Divide 30 by 2 to get 15.

$$10x=15$$

Divide both sides by 10.

$$x=\frac{15}{10}$$

Reduce the fraction \frac{15}{10} to lowest terms by extracting and canceling out 5.

$$x=\frac{3}{2}$$