Chapter 8, Problem 1RE

To determine

To sketch: The graph of the inequality $5x+2y\le 10$.

Explanation of Solution

In order to sketch the graph of the inequality $5x+2y\le 10$, first draw the straight line $5x+2y=10$ in the xy-plane.

Obtain the values of y, when $x=-2,0\text{ and 4}$ as in below table.

x	−2	0	4
y	10	5	−5

Since the given inequality is “less than or equal to”, the points on the line are part of the solution.

Hence, use a solid line to join the points in the above table and obtain the graph of $5x+2y=10$ as shown below in Figure 1.

From Figure 1, it is observed that the graph of the straight line $5x+2y=10$ intersects with the x and y axes at the points $\left(2,0\right)$ and $\left(0,5\right)$, respectively.

Take any arbitrary point as a test point which is not on the line $5x+2y=10$ to obtain the solution of the inequality $5x+2y\le 10$.

The half plane which contains the test point which satisfies the inequality is the solution.

Let $\left(0,0\right)$ be the test point from the left half plane.

Substitute the values $x=0$ and $y=0$ in the inequality $5x+2y\le 10$.

$\begin{array}{l}5\left(0\right)+2\left(0\right)\le 10\\ 0\le 10\end{array}$

This is true.

Therefore, the left half plane of the line $5x+2y=10$ satisfies the inequality $5x+2y\le 10$.

Use the above mentioned information and sketch the graph of the inequality $5x+2y\le 10$ as shown in the below Figure 2.

From Figure 2, it is observed that the shaded region and the line $5x+2y=10$ is the solution of the inequality $5x+2y\le 10$.