Solve 42x-3/3-1/2[1-x/3-1+x/2]>frac{9}{4}

$$4 \frac { 2 x - 3 } { 3 } - \frac { 1 } { 2 } [ \frac { 1 - x } { 3 } - \frac { 1 + x } { 2 } ] > \frac { 9 } { 4 }$$

Solve for x

$$x>2$$

Short Solution Steps

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

$$16\left(2x-3\right)-6\left(\frac{1-x}{3}-\frac{1+x}{2}\right)>27$$

Use the distributive property to multiply 16 by 2x-3.

$$32x-48-6\left(\frac{1-x}{3}-\frac{1+x}{2}\right)>27$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{1-x}{3} times \frac{2}{2}. Multiply \frac{1+x}{2} times \frac{3}{3}.

$$32x-48-6\left(\frac{2\left(1-x\right)}{6}-\frac{3\left(1+x\right)}{6}\right)>27$$

Since \frac{2\left(1-x\right)}{6} and \frac{3\left(1+x\right)}{6} have the same denominator, subtract them by subtracting their numerators.

$$32x-48-6\times \frac{2\left(1-x\right)-3\left(1+x\right)}{6}>27$$

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

$$32x-48-6\times \frac{2-2x-3-3x}{6}>27$$

Combine like terms in 2-2x-3-3x.

$$32x-48-6\times \frac{-1-5x}{6}>27$$

Express 6\times \frac{-1-5x}{6} as a single fraction.

$$32x-48-\frac{6\left(-1-5x\right)}{6}>27$$

Cancel out 6 and 6.

$$32x-48-\left(-1-5x\right)>27$$

To find the opposite of -1-5x, find the opposite of each term.

$$32x-48-\left(-1\right)-\left(-5x\right)>27$$

The opposite of -1 is 1.

$$32x-48+1-\left(-5x\right)>27$$

The opposite of -5x is 5x.

$$32x-48+1+5x>27$$

Add -48 and 1 to get -47.

$$32x-47+5x>27$$

Combine 32x and 5x to get 37x.

$$37x-47>27$$

Add 47 to both sides.

$$37x>27+47$$

Add 27 and 47 to get 74.

$$37x>74$$

Divide both sides by 37. Since 37 is positive, the inequality direction remains the same.

$$x>\frac{74}{37}$$

Divide 74 by 37 to get 2.

$$x>2$$