Solve 3left(2x-x+2/2right)geq6left(x+1right)+2x-4/3-x-10

$$3 \left( 2x- \frac{ x+2 }{ 2 } \right) \geq 6 \left( x+1 \right) + \frac{ 2x-4 }{ 3 } -x-10$$

Solve for x

$$x\leq 2$$

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.

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

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

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

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

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

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

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

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

$$27x-18\geq 36\left(x+1\right)+2\left(2x-4\right)-6x-60$$

Use the distributive property to multiply 36 by x+1.

$$27x-18\geq 36x+36+2\left(2x-4\right)-6x-60$$

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

$$27x-18\geq 36x+36+4x-8-6x-60$$

Combine 36x and 4x to get 40x.

$$27x-18\geq 40x+36-8-6x-60$$

Subtract 8 from 36 to get 28.

$$27x-18\geq 40x+28-6x-60$$

Combine 40x and -6x to get 34x.

$$27x-18\geq 34x+28-60$$

Subtract 60 from 28 to get -32.

$$27x-18\geq 34x-32$$

Subtract 34x from both sides.

$$27x-18-34x\geq -32$$

Combine 27x and -34x to get -7x.

$$-7x-18\geq -32$$

Add 18 to both sides.

$$-7x\geq -32+18$$

Add -32 and 18 to get -14.

$$-7x\geq -14$$

Divide both sides by -7. Since -7 is negative, the inequality direction is changed.

$$x\leq \frac{-14}{-7}$$

Divide -14 by -7 to get 2.

$$x\leq 2$$