Chapter 3.4, Problem 38STP

To determine

To calculate: The solution set of the given inequalities by graphing.

  $\begin{array}{c}3x+2y\ge 6\\ 4x-y\ge 2\end{array}$

Answer to Problem 38STP

The solution set of the inequalities $\begin{array}{c}3x+2y\ge 6\\ 4x-y\ge 2\end{array}$ is,

  

Explanation of Solution

Given information:

The inequalities $\begin{array}{c}3x+2y\ge 6\\ 4x-y\ge 2\end{array}$ .

Formula used:

To solve a system of inequalities by graphing, plot all the inequalities on the graph and shade the region that represents each inequality. The region that is common to all the inequalities i.e. shaded for all inequalities is the solution of the system.

Calculation:

Consider the given system of inequalities $\begin{array}{c}3x+2y\ge 6\\ 4x-y\ge 2\end{array}$ .

Recall that to solve a system of inequalities by graphing, plot all the inequalities on the graph and shade the region that represents each inequality. The region that is common to all the inequalities i.e. shaded for all inequalities is the solution of the system.

To know whether the region will be shaded towards origin or away from the origin, substitute $x=0$ and $y=0$ in the inequality. If inequality holds true region will be shaded towards origin and if not away from the origin.

Now, substitute $x=0$ and $y=0$ in the inequality $3x+2y\ge 6$ and check whether it is true or not and then plot on the graph.

  $\begin{array}{c}3x+2y\ge 6\\ 3\left(0\right)+2\left(0\right)\ge 6\\ 0\ge 6\end{array}$

Since, the inequality is not true, so, its region will be shaded away from the origin as,

  

Now, substitute $x=0$ and $y=0$ in the inequality $4x-y\ge 2$ and check whether it is true or not and then plot on the graph.

  $\begin{array}{c}4x-y\ge 2\\ 4\left(0\right)-\left(0\right)\ge 2\\ 0\ge 2\end{array}$

Since, the inequality is not true, so, its region will be shaded away from the origin as,

  

On combining both the inequalities on the graph, the common shaded region is unbounded region away from origin,

  

In the above figure, the color brown shaded region represents the solution set of the given system of inequalities.

Thus, the solution set of the inequalities $\begin{array}{c}3x+2y\ge 6\\ 4x-y\ge 2\end{array}$ is,