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Solving Systems of Linear Inequalities Graphically

Solving Systems of Linear Inequalities Graphically 1.10 - Solution

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a
There are two major steps to writing an inequality when given its graph.
  1. Write an equation for the boundary line
  2. Determine the inequality symbol and complete the inequality

In this exercise, we have been given a system consisting of two linear inequalities. We'll tackle them one at a time and bring them together in a system at the end. Let's call the inequalities and

Inequality

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've identified two points, and and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph. We can use this information to find the slope, One of the points we selected, is also the -intercept. With slope and the -intercept a we can write an equation for the boundary line in slope-intercept form. Now that we found the boundary line we should find the correct inequality symbol. To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation.

We'll substitute for this test, then make the inequality symbol fit the resulting statement.
Evaluate right-hand side
We know that is greater than 0, so the symbol will be either or Since the boundary line in the given graph is solid, the inequality is not strict. We can now form the first inequality in the system.

Inequality B

Writing the inequality for this region of the graph will involve the same steps as above. We'll start by identifying two points.

Again, we have denoted the rise and run of the graph, giving the slope Since we chose the -intercept at the point as one of our points for this boundary line as well, we can write its equation. Once more, we replace the equality sign with a question mark. To determine the correct inequaality symbol, we will use a test point.

We'll substitute for this test, then make the inequality symbol fit the resulting statement.
Simplify right-hand side
We see that the correct symbol should be or To chose between these two we'll look at the line and see that it's dashed. Thus, the correct symbol is To complete the system of inequalities, we will bring both of our inequalities together in system notation.
b
The two steps in writing an inequality when given its graph is to write equations of the boundary lines and then determine their inequality symbols. Let's do this for one equation at a time, where we name the line and

Inequality

First we need to identify two points on the line where one preferably is the -intercept.

Here we've identified two points, and 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, Now that we have the slope and the -intercept we can write an equation of the line in slope-intercept form. We should now determine the correct inequality symbol. This is done using a test point. Let's use the origin.

We'll substitute for this test, then make the inequality symbol fit the resulting statement.
Since is less than the symbol will be either or The boundary line in the given graph is dashed, indicating that the inequality is strict. Thus, the correct symbol is

Inequality

The boundary line of the second region of the graph is a vertical line. Therefore, writing an inequality for this region requires a different approach.

Notice that every point on the boundary line will have a -cordinate equal to This information is enough to write corresponding equation. Since the region to the left of the line is shaded it represents all -values less than We can now write our system of inequalities.