2.15.2 Determining if Points on Boundary Lines Are Solutions to a System

Extending Your Thinking

Find a system of inequalities with this triangle as its set of solutions:

$x+y\ge 2$, $x+4y\le 14$, $4x+y\le 11$ or equivalent

Determine Solutions of Systems of Inequalities on Boundary Lines

A solution to a system of inequalities only exists on a boundary line if the line is solid, not dashed.

Example

Solve the system by graphing:

$\{\begin{array}{c}x-y>3\hfill \\ y<-\frac{1}{5}x+4.\hfill \end{array}$

Solution

$\{\begin{array}{c}x-y>3\hfill \\ y<-\frac{1}{5}x+4\hfill \end{array}$

Graph $x-y>3$, by graphing $x-y=3$ and testing a point. The intercepts are $x=3$ and $y=-3$, and the boundary line will be dashed. Test $(0,0)$, which makes the inequality false, so shade (red) the side that does not contain $(0,0)$.

Graph $y<-\frac{1}{5}x+4$ by graphing $y=-\frac{1}{5}x+4$ using the slope $m=-15$ and $y$-intercept $b=4$. The boundary line will be dashed

Test $(0,0)$, which makes the inequality true, so shade (blue) the side that contains $(0,0)$.

Choose a test point in the solution and verify that it is a solution to both inequalities.

The point of intersection of the two lines is not included because both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.

Notice that the point $(0,-3)$ is NOT a solution because it falls on the red dashed line. The point $(4,-2)$ IS a solution because it is included in the solutions to both inequalities.

Try it

Try It: Determine Solutions of Systems of Inequalities on Boundary Lines

Which of the following are solutions of the graph below?

• $(2,1)$
• $(2,3)$
• $(5,0)$
• $(3,-1)$
• $(-1,6)$
• $(1,6)$

Compare your answer:

Here is how to determine if the coordinates are solutions:

Remember that when a coordinate is on a boundary line, it must be solid if it is a solution. The coordinate must also be in the overlapping shaded regions to be a solution to the entire system.

The answers are $(2,1)$ and $(3,-1)$