Solve x+3/2-1-x/3geq3-x/4+x-2/2

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

Solve for x

$$x\geq -\frac{17}{7}$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 2,3,4. Since 12 is positive, the inequality direction remains the same.

$$6\left(x+3\right)-4\left(1-x\right)\geq 3\left(3-x\right)+6\left(x-2\right)$$

Use the distributive property to multiply 6 by x+3.

$$6x+18-4\left(1-x\right)\geq 3\left(3-x\right)+6\left(x-2\right)$$

Use the distributive property to multiply -4 by 1-x.

$$6x+18-4+4x\geq 3\left(3-x\right)+6\left(x-2\right)$$

Subtract 4 from 18 to get 14.

$$6x+14+4x\geq 3\left(3-x\right)+6\left(x-2\right)$$

Combine 6x and 4x to get 10x.

$$10x+14\geq 3\left(3-x\right)+6\left(x-2\right)$$

Use the distributive property to multiply 3 by 3-x.

$$10x+14\geq 9-3x+6\left(x-2\right)$$

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

$$10x+14\geq 9-3x+6x-12$$

Combine -3x and 6x to get 3x.

$$10x+14\geq 9+3x-12$$

Subtract 12 from 9 to get -3.

$$10x+14\geq -3+3x$$

Subtract 3x from both sides.

$$10x+14-3x\geq -3$$

Combine 10x and -3x to get 7x.

$$7x+14\geq -3$$

Subtract 14 from both sides.

$$7x\geq -3-14$$

Subtract 14 from -3 to get -17.

$$7x\geq -17$$

Divide both sides by 7. Since 7 is positive, the inequality direction remains the same.

$$x\geq -\frac{17}{7}$$