Question

11: Solve each system of inequalities by graphing.

$\begin{array}{c}x+2y\ge 7\\ 3x-4y>12\end{array}$

Short answer

The solution is$7-2y\le x\le \frac{12+4y}{3}$.

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-:

$\begin{array}{c}x+2y\ge 7\\ 3x-4y<12\end{array}$

The linear equation of the inequalities is-:

$\begin{array}{c}x+2y=7\\ 3x-4y=12\end{array}$

The point which satisfies the equation $x+2y=7$ are $(7,0)$ and$(1,3)$.

The point which satisfies the equation $3x-4y=12$ are $(0,-3)$ and$(4,0)$.

Step-4 – Evaluating the shading region

We choose $(0,0)$then the point $(0,0)$ satisfies the inequality $3x-4y<12$ but it does not satisfy the inequality$x+2y\ge 7$.

Step-5 – Plotting the graph

So, the graph of the inequality is

The common region is$7-2y\le x\le \frac{12+4y}{3}$.