Graphing Inequalities on a Coordinate Plane

The first video explains the basics of graphing an inequality. The example is a word problem, in which the inequality is written in standard form.

 
The second video expands on the ideas above, using an inequality in slope-intercept form.

 
Graphing Inequalities on Coordinate Plane problem.

Try solving:     $3x+y \lt 5$

 
 
 
 

Solution:

1. The first step is to the coordinates that will create the boundary line (the boarder of the area that is represented by the inequality) by treating the inequality as an equality.
There are two ways to do this
        (a.) you could make a table of points for $\color{#500}{3x+y=5}$

$$
\begin{array}{c|c}
x & y \\
\hline
1 & 2 \\
\hline
0 & 5 \\
\hline
-1 & 8 \\
\hline
\end{array}
$$

        (b.) or you could rewrite the equation in slot-intercept form and use that information to graph the boundary line.
$$
\begin {align*}
3x+y &= 5\\
3x+\color{red}{^-3x}+y &= 5+\color{red}{^-3x} \\
y &= {^-3x}+5
\end{align*}
$$
So in this case, the slope = $\frac{-3}{1}$ and the y-intercept = $5$.
 

2. Instead of graphing a straight line we will need to use a dotted line, since the original inequality $3x+y \lt 5$ is less than not less than or equal to.

Boundary points for graph

3. To decide which side of the line to shade in on the graph, pick a coordinate pair on either side of the boundary line and see if it makes the inequality true. I usually use $(0,0)$ (unless it is on the boundary line) because it makes the math easier.

$$
\begin{align*}
3x+y &\lt 5 \\
3*\color{#f15a23}0+\color{#f15a23}0 &\lt 5 \\
0 &\lt 5
\end{align*}
$$
Because the statement is true we know that $(0,0)$ is part of the solution set, we would shade in the left hand side of the equation. The shaded area represents all possible solutions to the inequality $3x+y \lt 5$.
Solution to the inequality