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

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

Solve for x

$$x\leq 3$$

Solution Steps

Use the distributive property to multiply \frac{5}{6} by 3-x.

$$\frac{5}{6}\times 3+\frac{5}{6}\left(-1\right)x-\frac{1}{2}\left(x-4\right)\geq \frac{1}{2}$$

Express \frac{5}{6}\times 3 as a single fraction.

$$\frac{5\times 3}{6}+\frac{5}{6}\left(-1\right)x-\frac{1}{2}\left(x-4\right)\geq \frac{1}{2}$$

Multiply 5 and 3 to get 15.

$$\frac{15}{6}+\frac{5}{6}\left(-1\right)x-\frac{1}{2}\left(x-4\right)\geq \frac{1}{2}$$

Reduce the fraction \frac{15}{6} to lowest terms by extracting and canceling out 3.

$$\frac{5}{2}+\frac{5}{6}\left(-1\right)x-\frac{1}{2}\left(x-4\right)\geq \frac{1}{2}$$

Multiply \frac{5}{6} and -1 to get -\frac{5}{6}.

$$\frac{5}{2}-\frac{5}{6}x-\frac{1}{2}\left(x-4\right)\geq \frac{1}{2}$$

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

$$\frac{5}{2}-\frac{5}{6}x-\frac{1}{2}x-\frac{1}{2}\left(-4\right)\geq \frac{1}{2}$$

Express -\frac{1}{2}\left(-4\right) as a single fraction.

$$\frac{5}{2}-\frac{5}{6}x-\frac{1}{2}x+\frac{-\left(-4\right)}{2}\geq \frac{1}{2}$$

Multiply -1 and -4 to get 4.

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

Divide 4 by 2 to get 2.

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

Combine -\frac{5}{6}x and -\frac{1}{2}x to get -\frac{4}{3}x.

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

Convert 2 to fraction \frac{4}{2}.

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

Since \frac{5}{2} and \frac{4}{2} have the same denominator, add them by adding their numerators.

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

Add 5 and 4 to get 9.

$$\frac{9}{2}-\frac{4}{3}x\geq \frac{1}{2}$$

Subtract \frac{9}{2} from both sides.

$$-\frac{4}{3}x\geq \frac{1}{2}-\frac{9}{2}$$

Since \frac{1}{2} and \frac{9}{2} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{4}{3}x\geq \frac{1-9}{2}$$

Subtract 9 from 1 to get -8.

$$-\frac{4}{3}x\geq \frac{-8}{2}$$

Divide -8 by 2 to get -4.

$$-\frac{4}{3}x\geq -4$$

Multiply both sides by -\frac{3}{4}, the reciprocal of -\frac{4}{3}. Since -\frac{4}{3} is negative, the inequality direction is changed.

$$x\leq -4\left(-\frac{3}{4}\right)$$

Multiply -4 times -\frac{3}{4}.

$$x\leq 3$$