Solve 2(x-2/3)-3(1/2-2)>x/2

$$2 ( x - \frac { 2 } { 3 } ) - 3 ( \frac { 1 } { 2 } - 2 ) > \frac { x } { 2 }$$

Solve for x

$$x>-\frac{19}{9}$$

Solution Steps

Multiply both sides of the equation by 6, the least common multiple of 3,2. Since 6 is positive, the inequality direction remains the same.

$$12\left(x-\frac{2}{3}\right)-18\left(\frac{1}{2}-2\right)>3x$$

Use the distributive property to multiply 12 by x-\frac{2}{3}.

$$12x+12\left(-\frac{2}{3}\right)-18\left(\frac{1}{2}-2\right)>3x$$

Express 12\left(-\frac{2}{3}\right) as a single fraction.

$$12x+\frac{12\left(-2\right)}{3}-18\left(\frac{1}{2}-2\right)>3x$$

Multiply 12 and -2 to get -24.

$$12x+\frac{-24}{3}-18\left(\frac{1}{2}-2\right)>3x$$

Divide -24 by 3 to get -8.

$$12x-8-18\left(\frac{1}{2}-2\right)>3x$$

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

$$12x-8-18\left(\frac{1}{2}-\frac{4}{2}\right)>3x$$

Since \frac{1}{2} and \frac{4}{2} have the same denominator, subtract them by subtracting their numerators.

$$12x-8-18\times \frac{1-4}{2}>3x$$

Subtract 4 from 1 to get -3.

$$12x-8-18\left(-\frac{3}{2}\right)>3x$$

Express 18\left(-\frac{3}{2}\right) as a single fraction.

$$12x-8-\frac{18\left(-3\right)}{2}>3x$$

Multiply 18 and -3 to get -54.

$$12x-8-\frac{-54}{2}>3x$$

Divide -54 by 2 to get -27.

$$12x-8-\left(-27\right)>3x$$

The opposite of -27 is 27.

$$12x-8+27>3x$$

Add -8 and 27 to get 19.

$$12x+19>3x$$

Subtract 3x from both sides.

$$12x+19-3x>0$$

Combine 12x and -3x to get 9x.

$$9x+19>0$$

Subtract 19 from both sides. Anything subtracted from zero gives its negation.

$$9x>-19$$

Divide both sides by 9. Since 9 is positive, the inequality direction remains the same.

$$x>-\frac{19}{9}$$