Question

12: Solve each system of inequalities by graphing.

$\begin{array}{c}3x+y<-5\\ 2x-4y\ge 6\end{array}$

Short answer

The solution of the inequalities is$2y+3\le x<\frac{-5-y}{3}$.

Step-by-step solution

Step-1 – Concept of solution of linear inequalities

The solution of linear inequalities can be obtained by changing the inequalities into equations and solving the linear equations to obtain a graph. Then the common shaded region is a solution of the inequalities.

Step-2 – Concept of shading the region of the inequalities

The shaded region obtained by choosing a point, if the point satisfies the inequality the region is along the point, if not satisfies the inequalities, then the shaded region is opposite to the point.

Step-3 – Solving the inequalities

The given inequalities are-:

$\begin{array}{c}3x+y<-5\\ 2x-4y\ge 6\end{array}$

The linear equation of the inequalities is-:

$\begin{array}{c}3x+y=-5\\ 2x-4y=6\end{array}$

The point which satisfies the equation $3x+y=-5$are $(0,5)$ and$(2,-1)$.

The point which satisfies the equation $2x-4y=6$are $(3,0)$ and$(1,-1)$.

Step-4 – Evaluating the shading region

We choose the point $(0,0)$. The point which $(0,0)$does not satisfy any of the inequalities $3x+y<-5$ and $2x-4y\ge 6$.

Step-5 – Plotting the graph

So, the graph of the inequality is

The common shaded region is$2y+3\le x<\frac{-5-y}{3}$.