Solve 2/5(2x-1/3-x+2/4-x-4/12)-frac{4}{3}>frac{x-1}{2}-frac{4-x}{3}

$$\frac { 2 } { 5 } ( \frac { 2 x - 1 } { 3 } - \frac { x + 2 } { 4 } - \frac { x - 4 } { 12 } ) - \frac { 4 } { 3 } > \frac { x - 1 } { 2 } - \frac { 4 - x } { 3 }$$

Solve for x

$$x<\frac{3}{7}$$

Short Solution Steps

Multiply both sides of the equation by 60, the least common multiple of 5,3,4,12,2. Since 60 is positive, the inequality direction remains the same.

$$24\left(\frac{2x-1}{3}-\frac{x+2}{4}-\frac{x-4}{12}\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{2x-1}{3} times \frac{4}{4}. Multiply \frac{x+2}{4} times \frac{3}{3}.

$$24\left(\frac{4\left(2x-1\right)}{12}-\frac{3\left(x+2\right)}{12}-\frac{x-4}{12}\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

Since \frac{4\left(2x-1\right)}{12} and \frac{3\left(x+2\right)}{12} have the same denominator, subtract them by subtracting their numerators.

$$24\left(\frac{4\left(2x-1\right)-3\left(x+2\right)}{12}-\frac{x-4}{12}\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

Do the multiplications in 4\left(2x-1\right)-3\left(x+2\right).

$$24\left(\frac{8x-4-3x-6}{12}-\frac{x-4}{12}\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

Combine like terms in 8x-4-3x-6.

$$24\left(\frac{5x-10}{12}-\frac{x-4}{12}\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

Since \frac{5x-10}{12} and \frac{x-4}{12} have the same denominator, subtract them by subtracting their numerators.

$$24\times \frac{5x-10-\left(x-4\right)}{12}-80>30\left(x-1\right)-20\left(4-x\right)$$

Do the multiplications in 5x-10-\left(x-4\right).

$$24\times \frac{5x-10-x+4}{12}-80>30\left(x-1\right)-20\left(4-x\right)$$

Combine like terms in 5x-10-x+4.

$$24\times \frac{4x-6}{12}-80>30\left(x-1\right)-20\left(4-x\right)$$

Cancel out 12, the greatest common factor in 24 and 12.

$$2\left(4x-6\right)-80>30\left(x-1\right)-20\left(4-x\right)$$

Use the distributive property to multiply 2 by 4x-6.

$$8x-12-80>30\left(x-1\right)-20\left(4-x\right)$$

Subtract 80 from -12 to get -92.

$$8x-92>30\left(x-1\right)-20\left(4-x\right)$$

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

$$8x-92>30x-30-20\left(4-x\right)$$

Use the distributive property to multiply -20 by 4-x.

$$8x-92>30x-30-80+20x$$

Subtract 80 from -30 to get -110.

$$8x-92>30x-110+20x$$

Combine 30x and 20x to get 50x.

$$8x-92>50x-110$$

Subtract 50x from both sides.

$$8x-92-50x>-110$$

Combine 8x and -50x to get -42x.

$$-42x-92>-110$$

Add 92 to both sides.

$$-42x>-110+92$$

Add -110 and 92 to get -18.

$$-42x>-18$$

Divide both sides by -42. Since -42 is negative, the inequality direction is changed.

$$x<\frac{-18}{-42}$$

Reduce the fraction \frac{-18}{-42} to lowest terms by extracting and canceling out -6.

$$x<\frac{3}{7}$$