Chapter 1.9, Problem 35E

Solution

Solve the inequality $x^2-6x<0$ by graphing technique.

The solution of the inequality is $0<x<6$ .

Given:

The inequality is $x^2-6x<0$ .

Graph:

The graph for the inequality $x^2-6x<0$ is shown below.

Make the sketch for the parabola $y=x^2-6x$ and use dotted lines for the graph because the inequality sign is $\left(<\right)$ in the inequality $x^2-6x<0$ .

Since the graph cuts $x-$ axis at points $\left(0,0\right)\text{and }\left(6,0\right)$ . So, any point such that $0<x<6$ if satisfy the inequality $x^2-6x<0$ , then it will be solution of the inequality.

  

Interpretation:

The graph for the inequality $x^2-6x<0$ can be obtained by translating the graph of $y=x^2$ , $3$ units towards right and $9$ units downwards. The graph is an upward dotted parabola with $x-$ intercepts $\left(0,0\right)\text{ and }\left(6,0\right)$ . And the shaded region lies inside the parabolic region. Therefore, the solution for the inequality $x^2-6x<0$ is all point lying inside the shaded region excluding the dotted curve.

Hence the solution of the inequality is $0<x<6$ .