We are asked to graph the given inequality. To do this, we will create a compound inequality by removing the absolute value.
|x-y|>5
In this case, the solution set is any number greater than 5 units away from the midpoint in both the positive and negative directions.
This gives us two cases, one where x-y is greater than 5 and one where x-y is less than -5.
Absolute Value Inequality:& |x-y|>5
First Case:& x-y>5
Second Case:& x-y<-5
The solution to this type of compound inequality is the overlap of the solution sets. To graph it, we will determine its boundary lines by replacing the inequality signs with equals signs and then writing the equations in slope-intercept form.
Inequality
Boundary line Equation
y=mx+b
x-y>5
x-y=5
y= 1x+( -5)
x-y<-5
x-y=-5
y= 1x+ 5
Now let's plot the boundary lines of our inequality. Notice that because the inequality is strict, so both lines will be dashed.
Before we can shade the solution set of this compound inequality, we need to determine if the solution set lies between our boundary lines or outside them. To do that, we will need a test point that does not lie on either boundary line but lies between them.
It looks like the point ( 0, 0) will be a good test point. We will substitute this point for x and y in the given inequality and simplify. If the substitution creates a true statement, we shade the region between the boundary lines. Otherwise, we shade the region outside our boundary lines.
Since we ended up with a false statement we will shade the region that does not contain our test point. It means that our solution set is the region outside our boundary lines.
