Solve 1/3x+1/2<2/3-5/3x

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

Solve for x

$$x<\frac{1}{12}$$

Solution Steps

Add \frac{5}{3}x to both sides.

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

Combine \frac{1}{3}x and \frac{5}{3}x to get 2x.

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

Subtract \frac{1}{2} from both sides.

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

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

$$2x<\frac{4}{6}-\frac{3}{6}$$

Since \frac{4}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

$$2x<\frac{4-3}{6}$$

Subtract 3 from 4 to get 1.

$$2x<\frac{1}{6}$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x<\frac{\frac{1}{6}}{2}$$

Express \frac{\frac{1}{6}}{2} as a single fraction.

$$x<\frac{1}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$x<\frac{1}{12}$$