Q3. - Algebra 2

Question

State which region is the solution of the following system of inequalities. 

$\begin{array}{c}a.y\ge x\\ y\le -x\\ b.y\le x\\ y\ge -x\\ c.y\ge x\\ y\ge -x\\ d.y\le x\\ y\le -x\end{array}$

   

Step-by-Step Solution

Answer

The regions in which each system of inequalities lie is provided below.

$\begin{array}{cc}\begin{array}{c}a.y\ge x\\ y\le -x\end{array}& \text{Region 4}\\ \begin{array}{c}b.y\le x\\ y\ge -x\end{array}& \text{Region 2}\\ \begin{array}{c}c.y\ge x\\ y\ge -x\end{array}& \text{Region 1}\\ \begin{array}{c}d.y\le x\\ y\le -x\end{array}& \text{Region 3}\end{array}$

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 – Take a point from Region 1

Take a point from each region and substitute it in the inequalities and verify whether it is true or false.

 

Take a point from region 1 that does not lie on boundary of lines say $\left(0,1\right)$.

Now, substitute $\left(0,1\right)$ in the first system of inequalities to verify.

$\begin{array}{c}a.y\ge x\\ y\le -x\end{array}$

Verify the point.

 $\begin{array}{c}1\ge 0\\ 1\le 0\end{array}$

This is not true so, system of inequalities does not lie in Region 1.

Now, substitute $\left(0,1\right)$ in the second system of inequalities to verify.

 $\begin{array}{c}b.y\le x\\ y\ge -x\end{array}$

Verify the point.

$\begin{array}{c}1\le 0\\ 1\ge 0\end{array}$

This is not true so, system of inequalities does not lie in Region 1.

Now, substitute $\left(0,1\right)$ in the third system of inequalities to verify.

$\begin{array}{c}c.y\ge x\\ y\ge -x\end{array}$

Verify the point.

 $\begin{array}{c}1\ge 0\\ 1\ge 0\end{array}$

This is true so, system of inequalities lie in Region 1.

Hence, the system of inequalities that lie in Region 1 is provided below.

 $\begin{array}{c}c.y\ge x\\ y\ge -x\end{array}$

Step-3 – Take a point from Region 2

Take a point from region 2 that does not lie on boundary of lines say $\left(1,0\right)$.

Now, substitute $\left(1,0\right)$ in the first system of inequalities to verify.

$\begin{array}{c}a.y\ge x\\ y\le -x\end{array}$

 Verify the point.

$\begin{array}{c}0\ge 1\\ 0\le -1\end{array}$

This is not true so, system of inequalities does not lie in Region 2.

Now, substitute $\left(1,0\right)$ in the second system of inequalities to verify.

$\begin{array}{c}b.y\le x\\ y\ge -x\end{array}$

Verify the point.

 $\begin{array}{c}0\le 1\\ 0\ge -1\end{array}$

This is true so, system of inequalities lie in Region 2.

Hence, the system of inequalities that lie in Region 2 is provided below.

 $\begin{array}{c}b.y\le x\\ y\ge -x\end{array}$

Step-4 – Take a point from Region 3

Take a point from region 3 that does not lie on boundary of lines say $\left(0,-1\right)$.

Now, substitute $\left(0,-1\right)$ in the first system of inequalities to verify.

$\begin{array}{c}a.y\ge x\\ y\le -x\end{array}$

Verify the point.

 $\begin{array}{c}-1\ge 0\\ -1\le 0\end{array}$

This is not true so, system of inequalities does not lie in Region 3.

Now, substitute $\left(0,-1\right)$ in the second system of inequalities to verify.

$\begin{array}{c}d.y\le x\\ y\le -x\end{array}$

Verify the point.

 $\begin{array}{c}-1\le 0\\ -1\le 0\end{array}$

This is true so, system of inequalities lie in Region 3.

Hence, the system of inequalities that lie in Region 3 is provided below.

 $\begin{array}{c}d.y\le x\\ y\le -x\end{array}$

Step-5 – Take a point from Region 4

Take a point from region 3 that does not lie on boundary of lines say $\left(-1,0\right)$.

Now, substitute $\left(-1,0\right)$ in the first system of inequalities to verify.

 $\begin{array}{c}a.y\ge x\\ y\le -x\end{array}$

Verify the point.

 $\begin{array}{c}0\ge -1\\ 0\le 1\end{array}$

This is true so, system of inequalities lie in Region 4.

Hence, the system of inequalities that lie in Region 4 is provided below.

 $\begin{array}{c}a.y\ge x\\ y\le -x\end{array}$