Chapter 5.3, Problem 53PPS

To determine

To solve the inequality by using the graphical calculator

$\frac{1}{3}\left(4x+3\right)\ge \frac{2}{3}x+2$

Answer to Problem 53PPS

The solution of the inequality is:

  $\left[\begin{array}{cc}\text{Solution}:& x\ge \frac{3}{2}\\ \text{Decimal}:& x\ge 1.5\\ \text{Interval}\text{Notation}:& \left[\frac{3}{2},\infty \right)\end{array}\right]$

Explanation of Solution

Given:

$\frac{1}{3}\left(4x+3\right)\ge \frac{2}{3}x+2$

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:

$\frac{1}{3}\left(4x+3\right)\ge \frac{2}{3}x+2$

Now, rewrite the inequality as:

$\begin{array}{c}\frac{1}{3}\left(4x+3\right)\ge \frac{2}{3}x+2\\ \frac{1}{3}\left(4x+3\right)-\frac{2}{3}x-2\ge 0\\ \frac{1}{3}\left(4x+3-2x-6\right)\ge 0\\ \frac{1}{3}\left(2x-3\right)\ge 0\end{array}$

Further solve,

$\begin{array}{c}\frac{1}{3}\left(2x-3\right)\ge 0\\ \frac{1}{3}>0\text{ so, }2x-3\ge 0\\ x-\frac{3}{2}\ge 0\\ x-1.5\ge 0\end{array}$

Solve the inequality by using the graph of x − 1.5 = 0, draw the graph of the line of x − 1.5 = 0and 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.

  $\begin{array}{c}\text{At }\left(0,0\right):x-1.5\ge 0\\ \text{ 0}-1.5\ge 0\left\{\text{Not Satisfied}\right\}\end{array}$

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

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

Hence, by the graph the solution set of the inequality is given as $\left[\begin{array}{cc}\text{Solution}:& x\ge \frac{3}{2}\\ \text{Decimal}:& x\ge 1.5\\ \text{Interval}\text{Notation}:& \left[\frac{3}{2},\infty \right)\end{array}\right]$ .