Solve 1/2x-1/6<1/3x+1/4

$$\frac { 1 } { 2 } x - \frac { 1 } { 6 } < \frac { 1 } { 3 } x + \frac { 1 } { 4 }$$

Solve for x

$$x<\frac{5}{2}$$

Solution Steps

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

$$\frac{1}{2}x-\frac{1}{6}-\frac{1}{3}x<\frac{1}{4}$$

Combine \frac{1}{2}x and -\frac{1}{3}x to get \frac{1}{6}x.

$$\frac{1}{6}x-\frac{1}{6}<\frac{1}{4}$$

Add \frac{1}{6} to both sides.

$$\frac{1}{6}x<\frac{1}{4}+\frac{1}{6}$$

Least common multiple of 4 and 6 is 12. Convert \frac{1}{4} and \frac{1}{6} to fractions with denominator 12.

$$\frac{1}{6}x<\frac{3}{12}+\frac{2}{12}$$

Since \frac{3}{12} and \frac{2}{12} have the same denominator, add them by adding their numerators.

$$\frac{1}{6}x<\frac{3+2}{12}$$

Add 3 and 2 to get 5.

$$\frac{1}{6}x<\frac{5}{12}$$

Multiply both sides by 6, the reciprocal of \frac{1}{6}. Since \frac{1}{6} is positive, the inequality direction remains the same.

$$x<\frac{5}{12}\times 6$$

Express \frac{5}{12}\times 6 as a single fraction.

$$x<\frac{5\times 6}{12}$$

Multiply 5 and 6 to get 30.

$$x<\frac{30}{12}$$

Reduce the fraction \frac{30}{12} to lowest terms by extracting and canceling out 6.

$$x<\frac{5}{2}$$