Solve 4/3(3x-2)-3(1/3x-1)<4/5(2x-1)-1/2

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

Solve for x

$$x<-\frac{7}{6}$$

Solution Steps

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

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

Cancel out 3 and 3.

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

Express \frac{4}{3}\left(-2\right) as a single fraction.

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

Multiply 4 and -2 to get -8.

$$4x+\frac{-8}{3}-3\left(\frac{1}{3}x-1\right)<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Fraction \frac{-8}{3} can be rewritten as -\frac{8}{3} by extracting the negative sign.

$$4x-\frac{8}{3}-3\left(\frac{1}{3}x-1\right)<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

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

$$4x-\frac{8}{3}-3\times \frac{1}{3}x+3<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Multiply -3 times \frac{1}{3}.

$$4x-\frac{8}{3}-x+3<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Combine 4x and -x to get 3x.

$$3x-\frac{8}{3}+3<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Convert 3 to fraction \frac{9}{3}.

$$3x-\frac{8}{3}+\frac{9}{3}<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Since -\frac{8}{3} and \frac{9}{3} have the same denominator, add them by adding their numerators.

$$3x+\frac{-8+9}{3}<\frac{4}{5}\left(2x-1\right)-\frac{1}{2}$$

Add -8 and 9 to get 1.

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

Use the distributive property to multiply \frac{4}{5} by 2x-1.

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

Express \frac{4}{5}\times 2 as a single fraction.

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

Multiply 4 and 2 to get 8.

$$3x+\frac{1}{3}<\frac{8}{5}x+\frac{4}{5}\left(-1\right)-\frac{1}{2}$$

Multiply \frac{4}{5} and -1 to get -\frac{4}{5}.

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

Least common multiple of 5 and 2 is 10. Convert -\frac{4}{5} and \frac{1}{2} to fractions with denominator 10.

$$3x+\frac{1}{3}<\frac{8}{5}x-\frac{8}{10}-\frac{5}{10}$$

Since -\frac{8}{10} and \frac{5}{10} have the same denominator, subtract them by subtracting their numerators.

$$3x+\frac{1}{3}<\frac{8}{5}x+\frac{-8-5}{10}$$

Subtract 5 from -8 to get -13.

$$3x+\frac{1}{3}<\frac{8}{5}x-\frac{13}{10}$$

Subtract \frac{8}{5}x from both sides.

$$3x+\frac{1}{3}-\frac{8}{5}x<-\frac{13}{10}$$

Combine 3x and -\frac{8}{5}x to get \frac{7}{5}x.

$$\frac{7}{5}x+\frac{1}{3}<-\frac{13}{10}$$

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

$$\frac{7}{5}x<-\frac{13}{10}-\frac{1}{3}$$

Least common multiple of 10 and 3 is 30. Convert -\frac{13}{10} and \frac{1}{3} to fractions with denominator 30.

$$\frac{7}{5}x<-\frac{39}{30}-\frac{10}{30}$$

Since -\frac{39}{30} and \frac{10}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7}{5}x<\frac{-39-10}{30}$$

Subtract 10 from -39 to get -49.

$$\frac{7}{5}x<-\frac{49}{30}$$

Multiply both sides by \frac{5}{7}, the reciprocal of \frac{7}{5}. Since \frac{7}{5} is positive, the inequality direction remains the same.

$$x<-\frac{49}{30}\times \frac{5}{7}$$

Multiply -\frac{49}{30} times \frac{5}{7} by multiplying numerator times numerator and denominator times denominator.

$$x<\frac{-49\times 5}{30\times 7}$$

Do the multiplications in the fraction \frac{-49\times 5}{30\times 7}.

$$x<\frac{-245}{210}$$

Reduce the fraction \frac{-245}{210} to lowest terms by extracting and canceling out 35.

$$x<-\frac{7}{6}$$