Q2. - Algebra 2

Question

Tell whether the following statement is true or false If false, give a counterexample. A system of two linear inequalities has either no points or infinitely many points in its solution. 

Step-by-Step Solution

Answer

The statement “A system of two linear inequalities has either no points or infinitely many points in its solution” is true.

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

Take one inequality of greater than and one of less than sign.

For example: Consider system of inequalities. 

 $\begin{array}{c}y\ge 2\\ x\le -3\end{array}$

Consider the inequality provided below.

$y\ge 2$

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

Therefore, the boundary line will be solid.

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

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

Therefore, the boundary line will be solid.

Step-3 – Graph the inequalities

Graph the inequalities $y\ge 2$ and $x\le -3$ on same plane and shade the region.

Draw the line $y=2$.

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.

$0\ge 2$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

Draw the line $x=-3$.

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.

$0\le -3$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

Thus, the shaded regions are provided below.

The region 1 and 3 corresponds to inequality $y\ge 2$.

The region 2 and 3 corresponds to inequality $x\le -3$.

Step-4 – Shade the common region

The region common to both the inequalities $y\ge 2$ and $x\le -3$ is region 3.

Hence, there are infinitely many solution to the system of inequalities.