Solve (x+5/2)(x-3)<(x+1)^2-6

$$( x + \frac { 5 } { 2 } ) ( x - 3 ) < ( x + 1 ) ^ { 2 } - 6$$

Solve for x

$$x>-1$$

Solution Steps

Use the distributive property to multiply x+\frac{5}{2} by x-3 and combine like terms.

$$x^{2}-\frac{1}{2}x-\frac{15}{2}<\left(x+1\right)^{2}-6$$

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

$$x^{2}-\frac{1}{2}x-\frac{15}{2}<x^{2}+2x+1-6$$

Subtract 6 from 1 to get -5.

$$x^{2}-\frac{1}{2}x-\frac{15}{2}<x^{2}+2x-5$$

Subtract x^{2} from both sides.

$$x^{2}-\frac{1}{2}x-\frac{15}{2}-x^{2}<2x-5$$

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

$$-\frac{1}{2}x-\frac{15}{2}<2x-5$$

Subtract 2x from both sides.

$$-\frac{1}{2}x-\frac{15}{2}-2x<-5$$

Combine -\frac{1}{2}x and -2x to get -\frac{5}{2}x.

$$-\frac{5}{2}x-\frac{15}{2}<-5$$

Add \frac{15}{2} to both sides.

$$-\frac{5}{2}x<-5+\frac{15}{2}$$

Add -5 and \frac{15}{2} to get \frac{5}{2}.

$$-\frac{5}{2}x<\frac{5}{2}$$

Multiply both sides by -\frac{2}{5}, the reciprocal of -\frac{5}{2}. Since -\frac{5}{2} is negative, the inequality direction is changed.

$$x>\frac{5}{2}\left(-\frac{2}{5}\right)$$

Multiply \frac{5}{2} and -\frac{2}{5} to get -1.

$$x>-1$$