Begin by writing an equation for the boundary lines. If below the boundary line is shaded the inequality symbol will be ≤. If above the boundary line is shaded, the inequality symbol will be ≥.
Example System:y ≤ -1/2x+4 y ≥ 2x-6 y ≥ -6x-6
Practice makes perfect
For the given boundary lines, we can immediately determine their y-intercepts and with this we can find their slopes. Let's first indicate their y-intercepts!
Next, we will indicate another point on each line to find their slopes.
Let's first find the slope of the line that passes through the points (0,4) and (4,2) using the Slope Formula.
Now that we know the slope and y-intercept of the line, we can write its equation in slope-intercept form.
Boundary Line I
y= -1/2x+ 4
The equations for the other two lines can be written proceeding in the same way.
Boundary Line
Points
m=y_2-y_1/x_2-x_1
Slope
y-intercept
Equation
I:
( 0, 4),( 4, 2)
m=2- 4/4- 0
-1/2
( 0, 4)
y= -1/2x+ 4
II:
( 0, -6),( 4, 2)
m=2-( -6)/4- 0
2
( 0, -6)
y= 2x -6
III:
( 0, -6),( -2, 0)
m=0-( -6)/-2- 0
-3
( 0, -6)
y= -6x -6
In our last step, we will decide the inequality symbols. Since the boundary lines are in slope-intercept form, we have a special way to decide the symbols. If below the boundary line is shaded the inequality symbol will be ≤. If above the boundary line is shaded, the inequality symbol will be ≥.
ccc
& &Boundary Line &Inequality [0.5em]
&I: &y = -1/2x+4 &y ≤ -1/2x+4
&II: &y =2x-6 &y ≥ 2x-6
&III: & y = -6x-6 & y ≥ -6x-6
Finally, we have three inequalities to write a system.
y ≤ -1/2x+4 y ≥ 2x-6 y ≥ -6x-6
