Chapter 1.9, Problem 37E

Solution

Solve the inequality $x^2-4x+2>0$ by graphing technique.

The solution of the inequality is $x\in \left(-\infty ,0.586\right)\cup \left(3.414,\infty \right)$ .

Given:

The inequality is $x^2-4x+2>0$ .

Graph:

The graph for the inequality $x^2-4x+2>0$ is shown below.

Make the sketch for the parabola $y=x^2-4x+2$ and use dotted lines for the graph because the inequality sign is $\left(<\right)$ in the inequality $x^2-4x+2>0$ .

  

Interpretation:

From the graph it can be interpreted that the graph is an upward dotted parabola with $x-$ intercepts $\left(0.586,0\right)\text{ and }\left(3.414,0\right)$ . And the shaded region lies outside the parabolic region. Therefore, the solution for the inequality $x^2-4x+2>0$ is all point lying outside the shaded region excluding the dotted curve and $x-$ intercepts $\left(0.586,0\right)\text{ and }\left(3.414,0\right)$ , because of $\left(<\right)$ sign of inequality.

Hence the solution of the inequality is $x\in \left(-\infty ,0.586\right)\cup \left(3.414,\infty \right)$ .