Solve f(x)=-1/2(x-2)+2(-1/4x-1/3)-1/6

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

Evaluate

$$\frac{1}{6}-x$$

Solution Steps

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

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

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

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

Multiply -1 and -2 to get 2.

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

Divide 2 by 2 to get 1.

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

Use the distributive property to multiply 2 by -\frac{1}{4}x-\frac{1}{3}.

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

Express 2\left(-\frac{1}{4}\right) as a single fraction.

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

Multiply 2 and -1 to get -2.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Express 2\left(-\frac{1}{3}\right) as a single fraction.

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

Multiply 2 and -1 to get -2.

$$-\frac{1}{2}x+1-\frac{1}{2}x+\frac{-2}{3}-\frac{1}{6}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$-\frac{1}{2}x+1-\frac{1}{2}x-\frac{2}{3}-\frac{1}{6}$$

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

$$-x+1-\frac{2}{3}-\frac{1}{6}$$

Convert 1 to fraction \frac{3}{3}.

$$-x+\frac{3}{3}-\frac{2}{3}-\frac{1}{6}$$

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

$$-x+\frac{3-2}{3}-\frac{1}{6}$$

Subtract 2 from 3 to get 1.

$$-x+\frac{1}{3}-\frac{1}{6}$$

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

$$-x+\frac{2}{6}-\frac{1}{6}$$

Since \frac{2}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$-x+\frac{2-1}{6}$$

Subtract 1 from 2 to get 1.

$$-x+\frac{1}{6}$$

Expand

$$\frac{1}{6}-x$$

Solution Steps

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

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

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

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

Multiply -1 and -2 to get 2.

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

Divide 2 by 2 to get 1.

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

Use the distributive property to multiply 2 by -\frac{1}{4}x-\frac{1}{3}.

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

Express 2\left(-\frac{1}{4}\right) as a single fraction.

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

Multiply 2 and -1 to get -2.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

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

Express 2\left(-\frac{1}{3}\right) as a single fraction.

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

Multiply 2 and -1 to get -2.

$$-\frac{1}{2}x+1-\frac{1}{2}x+\frac{-2}{3}-\frac{1}{6}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$-\frac{1}{2}x+1-\frac{1}{2}x-\frac{2}{3}-\frac{1}{6}$$

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

$$-x+1-\frac{2}{3}-\frac{1}{6}$$

Convert 1 to fraction \frac{3}{3}.

$$-x+\frac{3}{3}-\frac{2}{3}-\frac{1}{6}$$

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

$$-x+\frac{3-2}{3}-\frac{1}{6}$$

Subtract 2 from 3 to get 1.

$$-x+\frac{1}{3}-\frac{1}{6}$$

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

$$-x+\frac{2}{6}-\frac{1}{6}$$

Since \frac{2}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$-x+\frac{2-1}{6}$$

Subtract 1 from 2 to get 1.

$$-x+\frac{1}{6}$$