Solve 1/2(3x-1)>(5x-2)+1/4

$$\frac { 1 } { 2 } ( 3 x - 1 ) > ( 5 x - 2 ) + \frac { 1 } { 4 }$$

Solve for x

$$x<\frac{5}{14}$$

Solution Steps

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

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{3}{2}x+\frac{1}{2}\left(-1\right)>5x-2+\frac{1}{4}$$

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

$$\frac{3}{2}x-\frac{1}{2}>5x-2+\frac{1}{4}$$

Convert -2 to fraction -\frac{8}{4}.

$$\frac{3}{2}x-\frac{1}{2}>5x-\frac{8}{4}+\frac{1}{4}$$

Since -\frac{8}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{3}{2}x-\frac{1}{2}>5x+\frac{-8+1}{4}$$

Add -8 and 1 to get -7.

$$\frac{3}{2}x-\frac{1}{2}>5x-\frac{7}{4}$$

Subtract 5x from both sides.

$$\frac{3}{2}x-\frac{1}{2}-5x>-\frac{7}{4}$$

Combine \frac{3}{2}x and -5x to get -\frac{7}{2}x.

$$-\frac{7}{2}x-\frac{1}{2}>-\frac{7}{4}$$

Add \frac{1}{2} to both sides.

$$-\frac{7}{2}x>-\frac{7}{4}+\frac{1}{2}$$

Least common multiple of 4 and 2 is 4. Convert -\frac{7}{4} and \frac{1}{2} to fractions with denominator 4.

$$-\frac{7}{2}x>-\frac{7}{4}+\frac{2}{4}$$

Since -\frac{7}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

$$-\frac{7}{2}x>\frac{-7+2}{4}$$

Add -7 and 2 to get -5.

$$-\frac{7}{2}x>-\frac{5}{4}$$

Multiply both sides by -\frac{2}{7}, the reciprocal of -\frac{7}{2}. Since -\frac{7}{2} is negative, the inequality direction is changed.

$$x<-\frac{5}{4}\left(-\frac{2}{7}\right)$$

Multiply -\frac{5}{4} times -\frac{2}{7} by multiplying numerator times numerator and denominator times denominator.

$$x<\frac{-5\left(-2\right)}{4\times 7}$$

Do the multiplications in the fraction \frac{-5\left(-2\right)}{4\times 7}.

$$x<\frac{10}{28}$$

Reduce the fraction \frac{10}{28} to lowest terms by extracting and canceling out 2.

$$x<\frac{5}{14}$$