Chapter 5.3, Problem 50PPS

To determine

To solve the inequality by using the graphical calculator

  $2\left(x-3\right)<3\left(2x+2\right)$

Answer to Problem 50PPS

The solution of the inequality is

  $\left[\begin{array}{cc}\text{Solution}:& x>-3\\ \text{Interval}\text{Notation}:& \left(-3,\infty \right)\end{array}\right]$ .

Explanation of Solution

Given:

  $2\left(x-3\right)<3\left(2x+2\right)$

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the dashed line at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the solid line at that endpoint.
• Properties of inequality:

1. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

To solve the inequality:

  $2\left(x-3\right)<3\left(2x+2\right)$

Now, expand 2(x -3) and 3(2x +2):

  $\begin{array}{c}2\left(x-3\right)<3\left(2x+2\right)\\ 2x-6<6x+6\\ 2x-6-6x-6<0\\ -4x-12<0\end{array}$

Solve the inequality by using the graph of −4 x + 12 = 0, draw the graph of the line −4 x + 12 = 0 and the plane divided into two parts.

Then, chose the origin point (0, 0) and put the point into the inequality to show weather it satisfied the inequality of not.

  $\begin{array}{c}\text{At }\left(0,0\right):-4x-12<0\\ -4\left(0\right)-12<0\left\{\text{ satisfied}\right\}\end{array}$

The point (0, 0) is in the solution set. So, the other plane is the solution set and the point that contains on the line are not part of the solution set. Thus, it is shown as a dashed line.

By using the Desmos graphical calculator, graph the following function as:

  

Hence, by the graph the solution of the inequality is

  $\left[\begin{array}{cc}\text{Solution}:& x>-3\\ \text{Interval}\text{Notation}:& \left(-3,\infty \right)\end{array}\right]$ .