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 should first draw each inequality separately. Then we will combine the graphs. The overlapping region will be the solution set. Let's start!
Inequality I
To determine the boundary line of the first inequality, we need to exchange the inequality symbol for an equals sign.
Inequality:& y ≥ |6-x|
Boundary Line:& y = |6-x|
The graph of this boundary line is the graph of the parent function y=|x| after transformations. Before we can identify them we have to rewrite our equation.
y=|6-x| ⇔ y=|x - 6|
Now we can identify a horizontal translation 6 units in the negative direction. The boundary line will be solid because the inequality is notstrict.
Next, we need decide which side of the boundary line we should shade. We can do this by testing a point that does not lie on the boundary line. If the point satisfies the inequality, it lies in the solution set. If not, we will shade the other region. Let's use ( 0, 0).
Because (0,0) did not create a true statement, we will shade the region that does not contain this point.
Inequality II
To graph |y|≤ 4, we will have to create a compound inequality first. Since |y| is less than or equal to x, this will be an and compound inequality.
|y|≤ 4 ⇔ y≤ 4 and y≥- 4
We can determine the boundary lines of this compound inequality by exchanging the inequality symbols to equal signs.
Boundary Lines: y=4 andy=- 4
Both boundary lines are horizontal. Since the inequalities are not strict the lines will be solid. The solution set of this inequality contains all coordinate pairs whose y-value is less than or equal to 4 andgreater than or equal to - 4. This means that we should shade the region between the lines.
Combining the Inequality Graphs
Let's draw the graphs of the inequalities on the same coordinate plane.
Finally, we can view only the solution set by removing the shaded regions that are not overlapping. Since the line y=- 4 is not a border of the overlapping region we will not include it in the final graph.
