Solve 6-x/5-3x+1/4+x-3/12leq3-3x/4-frac{5x-3}{2}

$$\frac{ 6-x }{ 5 } - \frac{ 3x+1 }{ 4 } + \frac{ x-3 }{ 12 } \leq \frac{ 3-3x }{ 4 } - \frac{ 5x-3 }{ 2 }$$

Solve for x

$$x\leq \frac{93}{143}$$

Solution Steps

Multiply both sides of the equation by 60, the least common multiple of 5,4,12,2. Since 60 is positive, the inequality direction remains the same.

$$12\left(6-x\right)-15\left(3x+1\right)+5\left(x-3\right)\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Use the distributive property to multiply 12 by 6-x.

$$72-12x-15\left(3x+1\right)+5\left(x-3\right)\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Use the distributive property to multiply -15 by 3x+1.

$$72-12x-45x-15+5\left(x-3\right)\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Combine -12x and -45x to get -57x.

$$72-57x-15+5\left(x-3\right)\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Subtract 15 from 72 to get 57.

$$57-57x+5\left(x-3\right)\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Use the distributive property to multiply 5 by x-3.

$$57-57x+5x-15\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Combine -57x and 5x to get -52x.

$$57-52x-15\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Subtract 15 from 57 to get 42.

$$42-52x\leq 15\left(3-3x\right)-30\left(5x-3\right)$$

Use the distributive property to multiply 15 by 3-3x.

$$42-52x\leq 45-45x-30\left(5x-3\right)$$

Use the distributive property to multiply -30 by 5x-3.

$$42-52x\leq 45-45x-150x+90$$

Combine -45x and -150x to get -195x.

$$42-52x\leq 45-195x+90$$

Add 45 and 90 to get 135.

$$42-52x\leq 135-195x$$

Add 195x to both sides.

$$42-52x+195x\leq 135$$

Combine -52x and 195x to get 143x.

$$42+143x\leq 135$$

Subtract 42 from both sides.

$$143x\leq 135-42$$

Subtract 42 from 135 to get 93.

$$143x\leq 93$$

Divide both sides by 143. Since 143 is positive, the inequality direction remains the same.

$$x\leq \frac{93}{143}$$