Graph each inequality separately. The overlapping region will be the solution of the system.
Practice makes perfect
To solve the given system by graphing, we will first graph each inequality separately, then combine the graphs. The overlapping region will be the solution of the system.
6y+2x≤ 9 & (I) 2y+18≥ 5x & (II) y>- 4x-9 & (III)
Graphing a single inequality involves two main steps.
Plotting the boundary line.
Shading half of the plane to show the solution set.
Let's begin!
Boundary Lines
We can tell a lot of information about the boundary lines from the inequalities given in the system.
Exchanging the inequality symbols for equals signs gives us the boundary line equations.
Observing the inequality symbols tells us whether the inequalities are strict.
Writing the equation in slope-intercept form will help us highlight the slopes m and y-intercepts b of the boundary lines.
Let's find each of these key pieces of information for the inequalities in the system.
Information
Inequality (I)
Inequality (II)
Inequality (III)
Given Inequality
6y+2x≤9
2y+18≥5x
y>- 4x-9
Boundary Line Equation
6y+2x=9
2y+18=5x
y=- 4x-9
Solid or Dashed?
≤ ⇒ Solid
≥ ⇒ Solid
> ⇒ Dashed
y= mx+ b
y= -1/3x+ 3/2
y= 5/2x+( - 9)
y= - 4x+( - 9)
Great! With all of this information, we can plot the boundary lines. Let's do it one at a time.
Inequality I
To draw the first boundary line we will plot the y-intercept and then use the slope to find another point.
To complete the graph, we will test a point that is not on the line and decide which region we should shade. Let's test the point ( 0, 0). If the point satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.
