Solve 2x-1/2-x-2/3leq2x-4/3-2x-2/12

$$\frac { 2 x - 1 } { 2 } - \frac { x - 2 } { 3 } \leq \frac { 2 x - 4 } { 3 } - \frac { 2 x - 2 } { 12 }$$

Solve for x

$$x\leq -8$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 2,3,12. Since 12 is positive, the inequality direction remains the same.

$$6\left(2x-1\right)-4\left(x-2\right)\leq 4\left(2x-4\right)-\left(2x-2\right)$$

Use the distributive property to multiply 6 by 2x-1.

$$12x-6-4\left(x-2\right)\leq 4\left(2x-4\right)-\left(2x-2\right)$$

Use the distributive property to multiply -4 by x-2.

$$12x-6-4x+8\leq 4\left(2x-4\right)-\left(2x-2\right)$$

Combine 12x and -4x to get 8x.

$$8x-6+8\leq 4\left(2x-4\right)-\left(2x-2\right)$$

Add -6 and 8 to get 2.

$$8x+2\leq 4\left(2x-4\right)-\left(2x-2\right)$$

Use the distributive property to multiply 4 by 2x-4.

$$8x+2\leq 8x-16-\left(2x-2\right)$$

To find the opposite of 2x-2, find the opposite of each term.

$$8x+2\leq 8x-16-2x-\left(-2\right)$$

The opposite of -2 is 2.

$$8x+2\leq 8x-16-2x+2$$

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

$$8x+2\leq 6x-16+2$$

Add -16 and 2 to get -14.

$$8x+2\leq 6x-14$$

Subtract 6x from both sides.

$$8x+2-6x\leq -14$$

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

$$2x+2\leq -14$$

Subtract 2 from both sides.

$$2x\leq -14-2$$

Subtract 2 from -14 to get -16.

$$2x\leq -16$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x\leq \frac{-16}{2}$$

Divide -16 by 2 to get -8.

$$x\leq -8$$