Solve 3(2x-x+2/2)geq5(x+1)+2x-4/3-x-10

$$3 ( 2 x - \frac { x + 2 } { 2 } ) \geq 5 ( x + 1 ) + \frac { 2 x - 4 } { 3 } - x - 10$$

Solve for x

$$x\leq 20$$

Solution Steps

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

$$18\left(2x-\frac{x+2}{2}\right)\geq 30\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Use the distributive property to multiply 18 by 2x-\frac{x+2}{2}.

$$36x+18\left(-\frac{x+2}{2}\right)\geq 30\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Cancel out 2, the greatest common factor in 18 and 2.

$$36x-9\left(x+2\right)\geq 30\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Use the distributive property to multiply -9 by x+2.

$$36x-9x-18\geq 30\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Combine 36x and -9x to get 27x.

$$27x-18\geq 30\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Use the distributive property to multiply 30 by x+1.

$$27x-18\geq 30x+30+2\left(2x-4\right)-6x-60$$

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

$$27x-18\geq 30x+30+4x-8-6x-60$$

Combine 30x and 4x to get 34x.

$$27x-18\geq 34x+30-8-6x-60$$

Subtract 8 from 30 to get 22.

$$27x-18\geq 34x+22-6x-60$$

Combine 34x and -6x to get 28x.

$$27x-18\geq 28x+22-60$$

Subtract 60 from 22 to get -38.

$$27x-18\geq 28x-38$$

Subtract 28x from both sides.

$$27x-18-28x\geq -38$$

Combine 27x and -28x to get -x.

$$-x-18\geq -38$$

Add 18 to both sides.

$$-x\geq -38+18$$

Add -38 and 18 to get -20.

$$-x\geq -20$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x\leq \frac{-20}{-1}$$

Fraction \frac{-20}{-1} can be simplified to 20 by removing the negative sign from both the numerator and the denominator.

$$x\leq 20$$