Question

Solve each inequality analyticaly. Support answers graphically. x^2+5x<2

Answer 1

It is given that:

`\begin{gathered} x^2+5x<2 \\ x^2+5x-2<0 \end{gathered}`

Solve for x to get:

`\begin{gathered} x^2+5x+(25)/(4)-2-(25)/(4)<0 \\ (x+(5)/(2))^2-((√(33))/(2))^2<0 \\ (x+\frac{5+\sqrt[]{33}}{2})(x+\frac{5-\sqrt[]{33}}{2})<0 \end{gathered}`

The two values are less than 0 if either one of them is negative so it follows:

`\begin{gathered} x+\frac{5+\sqrt[]{33}}{2}<0\text{ and }x+\frac{5-\sqrt[]{33}}{2}>0 \\ OR \\ x+\frac{5+\sqrt[]{33}}{2}>0\text{ and }x+\frac{5-\sqrt[]{33}}{2}<0 \end{gathered}`

The graph is given as below:

The points are x=-5.372 and x=0.372 which is the same as:

`x=-\frac{5+\sqrt[]{33}}{2}=-5.372,x=-\frac{5-\sqrt[]{33}}{2}=-0.372`