Determine the inequality symbol and complete the inequality.
In this exercise we have been given a system 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, (-1,0) 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=1/1 ⇔ m= 1One 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 slope-intercept form.
y= mx+ b ⇒ y= 1x+ 1
To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space, since the symbol is still unknown to us.
y ? x+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 ( -1, 1) for this test, then make the inequality symbol fit the resulting statement.
Since 1 is greater than 0, the symbol will be either > or ≥. The boundary line in the given graph is dashed, so the inequality is strict. We can now form the first inequality in the system.
y > x+1
The Blue Region
Since the boundary line of the second region of the graph is a horizontal line, writing the inequality for this region requires a different approach.
Notice that the y-coordinate of every point on the boundary line is equal to 2. This information is enough to write the corresponding equation.
y=2
Once more, we replace the equals sign with a blank space.
y ? 2
We will need a point that lies within the solution set to determine the sign of this inequality.
We will substitute ( 1, 3) for this test, then make the inequality symbol fit the resulting statement.
Since 3 is greater than 2, the symbol will be either > or ≥. The boundary line in the given graph is solid, so the inequality is non-strict. We can now form the second inequality in the system.
y≥ 2
Writing the System
To complete the system of inequalities, we will bring both of our inequalities together in system notation.
y >x+1 y≥2
