Solve 4(2x-1)-x+(2x-1)^2>4(x-2)^2-12(x-1)

$$4 ( 2 x - 1 ) - x + ( 2 x - 1 ) ^ { 2 } > 4 ( x - 2 ) ^ { 2 } - 12 ( x - 1 )$$

Solve for x

$$x>1$$

Solution Steps

Use the distributive property to multiply 4 by 2x-1.

$$8x-4-x+\left(2x-1\right)^{2}>4\left(x-2\right)^{2}-12\left(x-1\right)$$

Combine 8x and -x to get 7x.

$$7x-4+\left(2x-1\right)^{2}>4\left(x-2\right)^{2}-12\left(x-1\right)$$

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

$$7x-4+4x^{2}-4x+1>4\left(x-2\right)^{2}-12\left(x-1\right)$$

Combine 7x and -4x to get 3x.

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

Add -4 and 1 to get -3.

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

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

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

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

$$3x-3+4x^{2}>4x^{2}-16x+16-12\left(x-1\right)$$

Use the distributive property to multiply -12 by x-1.

$$3x-3+4x^{2}>4x^{2}-16x+16-12x+12$$

Combine -16x and -12x to get -28x.

$$3x-3+4x^{2}>4x^{2}-28x+16+12$$

Add 16 and 12 to get 28.

$$3x-3+4x^{2}>4x^{2}-28x+28$$

Subtract 4x^{2} from both sides.

$$3x-3+4x^{2}-4x^{2}>-28x+28$$

Combine 4x^{2} and -4x^{2} to get 0.

$$3x-3>-28x+28$$

Add 28x to both sides.

$$3x-3+28x>28$$

Combine 3x and 28x to get 31x.

$$31x-3>28$$

Add 3 to both sides.

$$31x>28+3$$

Add 28 and 3 to get 31.

$$31x>31$$

Divide both sides by 31. Since 31 is positive, the inequality direction remains the same.

$$x>\frac{31}{31}$$

Divide 31 by 31 to get 1.

$$x>1$$