Find the regions or solution for the following inequations:

a) $$x^2+x+1 < -x$$

b) $$x(1-x)+2x-2 > -1$$

c) $$(x-2)(x+1)+x > (2x-1)x -2$$

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### Development:

a) $$x^2+x+1 < -x \Rightarrow x^2+2x+1 < 0$$

We find the solutions of the equation $$x^2+2x+1=0$$: $$x=\dfrac{-2\pm\sqrt{4-4}}{2}=-1$$$There is only one solution. So, $$x^2+2x+1 < 0 \Rightarrow (x-1)^2<0$$$ There is no solution, because the square of a number is always positive.

b) $$x(1-x)+2x-2 > -1 \Rightarrow x-x^2 +2x-2+1 > 0 \Rightarrow$$$$$\Rightarrow -x^2+3x-1 > 0 \Rightarrow x^2-3x+1 < 0$$$

We find the solutions of the equation $$x^2-3x+1=0$$: $$x=\dfrac{ 3\pm\sqrt{9-4}}{2}=\dfrac{3\pm\sqrt{5}}{2} \Rightarrow x_1= \dfrac{3-\sqrt{5}}{2}, \ x_2=\dfrac{3+\sqrt{5}}{2}$$$where $$x_1 < x_2$$. $$x^2-3x+1 < 0 \Rightarrow (x-x_1)(x-x_2) < 0 \Rightarrow$$$ $$\Rightarrow \left\{ \begin{array}{l} \text{a) } \ (x-x_1) > 0 \ \text{ and } \ (x-x_2) < 0 \\ \text{b) } \ (x-x_1) < 0 \ \text{ and } \ (x-x_2) > 0 \end{array} \right.$$$$$\Rightarrow \left\{ \begin{array}{l} \text{a) } \ x > x_1 \ \text{ and } \ x < x_2 \\ \text{b) } \ x < x_1 \ \text{ and } \ x > x_2 \end{array} \right.$$$

and as $$x_1 < x_2$$, we have $$x_1 < x < x_2$$.

c) $$(x-2)(x+1)+x > (2x-1)x -2 \ \Rightarrow \ x^2+x-2x-2+x > 2x^2-x -2$$$$$\Rightarrow \ 0 > 2x^2-x-2-x^2 -x +2x +2 -x \ \Rightarrow \ 0 > x^2 -x$$$

We find the solutions of the equation $$x^2 -x=0$$: $$0=x^2-x=x(x-1) \Rightarrow x_1=0 \ \text{ and } \ (x-1)=0 \Rightarrow x_2=1$$$We have two solutions: $$0$$ and $$1$$. $$x^2 -x < 0 \Rightarrow x(x-1) < 0 \Rightarrow$$$ $$\Rightarrow \left\{ \begin{array}{l} x > 0 \ \text{ and } \ x-1 < 0 \Rightarrow x < 1\\ x < 0 \ \text{ and } \ x-1 > 0 \Rightarrow x > 1 \end{array} \right.$$\$

by more restrictive inequations we obtain $$x > 0$$ and $$x<1$$.

### Solution:

a) There is no solution.

b) The inequation is satisfied for those $$x$$ such that: $$\dfrac{3-\sqrt{5}}{2} < x < \dfrac{3+\sqrt{5}}{2}$$.

c) The inequation is satisfied for those $$x$$ such that: $$0 < x < 1$$.

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