There are two major steps to writing an when given its .
- Write an equation for the .
- Determine the inequality symbol and complete the inequality.
We have been given a consisting of two linear inequalities. We will tackle them one at a time and bring them together in a system at the end.
The Yellow Region
It only takes two points to create a unique equation for any line, so let's start by identifying two points on the boundary line.
Here we have identified two points, (-3,3) and (0, 1), and indicated the horizontal and vertical changes between them. This gives us the
rise
and
run
of the graph, which will give us the slope m.
rise/run=- 2/3 ⇔ m= - 2/3
One of the points we selected, (0, 1), is also the y-intercept. With slope m and the y-intercept at the point (0, b), we can write an equation for the boundary line in .
y= mx+ b ⇒ y=( - 2/3)x+ 1
To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space, since it is still unknown to us.
y ? (-2/3)x+1 ⇒ y ? - 2/3x+1
To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation.
We will substitute ( -2, -2) for this test, then make the inequality symbol fit the resulting statement.
y ? - 2/3x+1
-2 ? -2/3( -2)+1
-2 ? 2/3* 2+1
-2 ? 4/3+1
-2 ? 4/3+3/3
-2 ? 4+3/3
-2 ? 7/3
The Blue Region
Writing the inequality for this region of the graph will involve the same steps as above. We will start by identifying two points.
Again, we have denoted the rise
and run
of the graph, giving the slope m.
rise/run=1/2 ⇔ m= 1/2
Since we chose the y-intercept at the point (0, 3) as one of our points for this boundary line as well, we can write its equation.
y= mx+ b ⇒ y= 1/2x+ 3
Once more, we replace the equals sign with a blank space.
y ? 1/2x+3
We will need another point that lies within the solution set to determine the sign of this inequality.
We will substitute ( -2, 3) for this test, then make the inequality symbol fit the resulting statement.
y ? 1/2x+3
3 ? 1/2( -2)+3
3 ? 2
Writing the System
To complete the system of inequalities, we will bring both of our inequalities together in system notation.
y ≤ - 23x+1 y > 12x+3
