Solve 2/3(x-2-x-1/2)geq1-2/3x-2/3(x-frac{x}{2})+2(frac{2}{3}x-1)

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

Solve for x

$$x\in \mathrm{R}$$

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.

$$4\left(x-2-\frac{x-1}{2}\right)\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

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

$$4x-8+4\left(-\frac{x-1}{2}\right)\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

Cancel out 2, the greatest common factor in 4 and 2.

$$4x-8-2\left(x-1\right)\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

Use the distributive property to multiply -2 by x-1.

$$4x-8-2x+2\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

Combine 4x and -2x to get 2x.

$$2x-8+2\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

Add -8 and 2 to get -6.

$$2x-6\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\left(x-\frac{x}{2}\right)\right)+12\left(\frac{2}{3}x-1\right)$$

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

$$2x-6\geq 6\left(1-\frac{2}{3}x-\frac{2}{3}\times \frac{1}{2}x\right)+12\left(\frac{2}{3}x-1\right)$$

Multiply \frac{2}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$2x-6\geq 6\left(1-\frac{2}{3}x-\frac{2\times 1}{3\times 2}x\right)+12\left(\frac{2}{3}x-1\right)$$

Cancel out 2 in both numerator and denominator.

$$2x-6\geq 6\left(1-\frac{2}{3}x-\frac{1}{3}x\right)+12\left(\frac{2}{3}x-1\right)$$

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

$$2x-6\geq 6\left(1-x\right)+12\left(\frac{2}{3}x-1\right)$$

Use the distributive property to multiply 6 by 1-x.

$$2x-6\geq 6-6x+12\left(\frac{2}{3}x-1\right)$$

Use the distributive property to multiply 12 by \frac{2}{3}x-1.

$$2x-6\geq 6-6x+12\times \frac{2}{3}x-12$$

Express 12\times \frac{2}{3} as a single fraction.

$$2x-6\geq 6-6x+\frac{12\times 2}{3}x-12$$

Multiply 12 and 2 to get 24.

$$2x-6\geq 6-6x+\frac{24}{3}x-12$$

Divide 24 by 3 to get 8.

$$2x-6\geq 6-6x+8x-12$$

Combine -6x and 8x to get 2x.

$$2x-6\geq 6+2x-12$$

Subtract 12 from 6 to get -6.

$$2x-6\geq -6+2x$$

Subtract 2x from both sides.

$$2x-6-2x\geq -6$$

Combine 2x and -2x to get 0.

$$-6\geq -6$$

This is true for any x.

$$x\in \mathrm{R}$$