Solve left(2x-1right)^2/4-left(3x-1right)left(3x+1right)/9<1/3-2x+3/12

$$\frac{ { \left(2x-1 \right) }^{ 2 } }{ 4 } - \frac{ \left( 3x-1 \right) \left( 3x+1 \right) }{ 9 } < \frac{ 1 }{ 3 } - \frac{ 2x+3 }{ 12 }$$

Solve for x

$$x>\frac{1}{3}$$

Solution Steps

Multiply both sides of the equation by 36, the least common multiple of 4,9,3,12. Since 36 is positive, the inequality direction remains the same.

$$9\left(2x-1\right)^{2}-4\left(3x-1\right)\left(3x+1\right)<12-3\left(2x+3\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$9\left(4x^{2}-4x+1\right)-4\left(3x-1\right)\left(3x+1\right)<12-3\left(2x+3\right)$$

Use the distributive property to multiply 9 by 4x^{2}-4x+1.

$$36x^{2}-36x+9-4\left(3x-1\right)\left(3x+1\right)<12-3\left(2x+3\right)$$

Use the distributive property to multiply -4 by 3x-1.

$$36x^{2}-36x+9+\left(-12x+4\right)\left(3x+1\right)<12-3\left(2x+3\right)$$

Use the distributive property to multiply -12x+4 by 3x+1 and combine like terms.

$$36x^{2}-36x+9-36x^{2}+4<12-3\left(2x+3\right)$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-36x+9+4<12-3\left(2x+3\right)$$

Add 9 and 4 to get 13.

$$-36x+13<12-3\left(2x+3\right)$$

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

$$-36x+13<12-6x-9$$

Subtract 9 from 12 to get 3.

$$-36x+13<3-6x$$

Add 6x to both sides.

$$-36x+13+6x<3$$

Combine -36x and 6x to get -30x.

$$-30x+13<3$$

Subtract 13 from both sides.

$$-30x<3-13$$

Subtract 13 from 3 to get -10.

$$-30x<-10$$

Divide both sides by -30. Since -30 is negative, the inequality direction is changed.

$$x>\frac{-10}{-30}$$

Reduce the fraction \frac{-10}{-30} to lowest terms by extracting and canceling out -10.

$$x>\frac{1}{3}$$