Solve (x(x+1)geq(x+2)(x-2)-2(x-1)

$$( x ( x + 1 ) \geq ( x + 2 ) ( x - 2 ) - 2 ( x - 1 )$$

Solve for x

$$x\geq -\frac{2}{3}$$

Solution Steps

Use the distributive property to multiply x by x+1.

$$x^{2}+x\geq \left(x+2\right)\left(x-2\right)-2\left(x-1\right)$$

Consider \left(x+2\right)\left(x-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$x^{2}+x\geq x^{2}-4-2\left(x-1\right)$$

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

$$x^{2}+x\geq x^{2}-4-2x+2$$

Add -4 and 2 to get -2.

$$x^{2}+x\geq x^{2}-2-2x$$

Subtract x^{2} from both sides.

$$x^{2}+x-x^{2}\geq -2-2x$$

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

$$x\geq -2-2x$$

Add 2x to both sides.

$$x+2x\geq -2$$

Combine x and 2x to get 3x.

$$3x\geq -2$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$x\geq -\frac{2}{3}$$