Solve 1/2x-3/4geq1/3x-5/6

$$\frac { 1 } { 2 } x - \frac { 3 } { 4 } \geq \frac { 1 } { 3 } x - \frac { 5 } { 6 }$$

Solve for x

$$x\geq -\frac{1}{2}$$

Solution Steps

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

$$\frac{1}{2}x-\frac{3}{4}-\frac{1}{3}x\geq -\frac{5}{6}$$

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

$$\frac{1}{6}x-\frac{3}{4}\geq -\frac{5}{6}$$

Add \frac{3}{4} to both sides.

$$\frac{1}{6}x\geq -\frac{5}{6}+\frac{3}{4}$$

Least common multiple of 6 and 4 is 12. Convert -\frac{5}{6} and \frac{3}{4} to fractions with denominator 12.

$$\frac{1}{6}x\geq -\frac{10}{12}+\frac{9}{12}$$

Since -\frac{10}{12} and \frac{9}{12} have the same denominator, add them by adding their numerators.

$$\frac{1}{6}x\geq \frac{-10+9}{12}$$

Add -10 and 9 to get -1.

$$\frac{1}{6}x\geq -\frac{1}{12}$$

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

$$x\geq -\frac{1}{12}\times 6$$

Express -\frac{1}{12}\times 6 as a single fraction.

$$x\geq \frac{-6}{12}$$

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

$$x\geq -\frac{1}{2}$$