Solve 1/2(2-x)>1/4(3-x)+1/4

$$\frac { 1 } { 2 } ( 2 - x ) > \frac { 1 } { 4 } ( 3 - x ) + \frac { 1 } { 4 }$$

Solve for x

$$x<0$$

Solution Steps

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

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

Cancel out 2 and 2.

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

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

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

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

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

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

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

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

$$1-\frac{1}{2}x>\frac{3}{4}-\frac{1}{4}x+\frac{1}{4}$$

Since \frac{3}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$1-\frac{1}{2}x>\frac{3+1}{4}-\frac{1}{4}x$$

Add 3 and 1 to get 4.

$$1-\frac{1}{2}x>\frac{4}{4}-\frac{1}{4}x$$

Divide 4 by 4 to get 1.

$$1-\frac{1}{2}x>1-\frac{1}{4}x$$

Add \frac{1}{4}x to both sides.

$$1-\frac{1}{2}x+\frac{1}{4}x>1$$

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

$$1-\frac{1}{4}x>1$$

Subtract 1 from both sides.

$$-\frac{1}{4}x>1-1$$

Subtract 1 from 1 to get 0.

$$-\frac{1}{4}x>0$$

Product of two numbers is >0 if both are >0 or both are <0. Since -\frac{1}{4}<0, x must be <0.

$$x<0$$