Solve 1/3(1-2x)<3/2(2x-1)

$$\frac{ 1 }{ 3 } (1-2x) < \frac{ 3 }{ 2 } (2x-1)$$

Solve for x

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

Solution Steps

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

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

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

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

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

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

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

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

Cancel out 2 and 2.

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

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

$$\frac{1}{3}-\frac{2}{3}x<3x-\frac{3}{2}$$

Subtract 3x from both sides.

$$\frac{1}{3}-\frac{2}{3}x-3x<-\frac{3}{2}$$

Combine -\frac{2}{3}x and -3x to get -\frac{11}{3}x.

$$\frac{1}{3}-\frac{11}{3}x<-\frac{3}{2}$$

Subtract \frac{1}{3} from both sides.

$$-\frac{11}{3}x<-\frac{3}{2}-\frac{1}{3}$$

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

$$-\frac{11}{3}x<-\frac{9}{6}-\frac{2}{6}$$

Since -\frac{9}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{11}{3}x<\frac{-9-2}{6}$$

Subtract 2 from -9 to get -11.

$$-\frac{11}{3}x<-\frac{11}{6}$$

Multiply both sides by -\frac{3}{11}, the reciprocal of -\frac{11}{3}. Since -\frac{11}{3} is negative, the inequality direction is changed.

$$x>-\frac{11}{6}\left(-\frac{3}{11}\right)$$

Multiply -\frac{11}{6} times -\frac{3}{11} by multiplying numerator times numerator and denominator times denominator.

$$x>\frac{-11\left(-3\right)}{6\times 11}$$

Do the multiplications in the fraction \frac{-11\left(-3\right)}{6\times 11}.

$$x>\frac{33}{66}$$

Reduce the fraction \frac{33}{66} to lowest terms by extracting and canceling out 33.

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