Question

Find the area of the region defined by system of inequalities $y+x\le 3,y-x\le 3\text{and}y\ge -1$.

Short answer

The area of the triangleis12 units square.

Step-by-step solution

Step-1 – Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid.

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality.

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-2 – Interpret the sign of the inequality

Consider the inequality provided below.

$y+x\le 3$

The inequality contains the sign of less than or equal to.

Therefore, the boundary line will be solid.

Next, consider the inequality $y-x\le 3$.

The inequality contains the sign of less than or equal to.

Therefore, the boundary line will be solid.

Next, consider the inequality $y\ge -1$.

The inequality contains the sign of greater than or equal to.

Therefore, the boundary line will be solid.

Step-3 – Graph the inequalities

Graph the inequalities $y+x\le 3$, $y-x\le 3$ and $y\ge -1$ on same plane and shade the region.

Draw the line.

$y+x=3$

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$.

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}0+0\le 3\\ 0\le 3\end{array}$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Draw the line.

$y-x=3$

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$.

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}0-0\le 3\\ 0\le 3\end{array}$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Draw the line $y=-1$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$.

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$0\ge -1$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Thus, the common shaded region is provided below.

Step-4 – Find the coordinates

The point of intersection of inequalities are the required coordinates of the figure.

The figure obtained by system of inequalities is a triangle.

The coordinates of vertices of the triangle are $\left(0,3\right)$, $\left(-4,-1\right)$ and $\left(4,-1\right)$.

Step-5 – Find the area

The area of the triangle is half times the base and height.

The base of the triangle is 8 units.

The height of the triangle is 3 units.

Therefore, the area of triangle is expressed as below.

$\begin{array}{c}A=\frac{1}{2}\left(\text{base}\right)\left(\text{height}\right)\\ =\frac{1}{2}\left(8\right)\left(3\right)\\ =12\end{array}$

Thus, the area of the triangle is 12 units square.