Graphing Linear Inequalities
After we are comfortable with solving basic inequalities and graphing linear equations, we can move on to solving linear inequalities in two variables and graphing regions. Solving linear equalities is just combining the concepts of inequalities and linear equations. If we think about how we graphed inequalities on a number line, it is a very similar process with linear equations.
Consider the inequality
If we recall how to solve this, we would isolate and solve for x
Then we would see that x is greater than -5, which means we would draw an open circle around -5 shade everything to the right of -5.
The steps for solving two variable linear inequalities are very similar. Let's try a couple examples.
(1) Graph the region of the linear inequality
Looking at this inequality, we should notice a number of things. It is in Slope-Intercept form (y = mx+b), which means we can identify the slope and y-intercept of this inequality. We will deal with the inequality sign once we have graphed the image. For now, we will treat it as an equal sign.
First, let's construct the graph. We can do this a number of ways, either by plugging in x values and obtaining their corresponding y values, or we could use the slope and y intercept in the inequality. Let's plot the y intercept and use the slope to form the line. we can see that b = 3, so the y intercept will be (0,3). The slope is m = -2, so we can go down 2 and right 1 (-2/1) or up 2 and left 1 (2/-1) to find the next point on the line.
We can also solve for these points on the line algebraically using the slope formula.
We know a point on the line (the y intercept) and the slope, so we can solve for another point.
We have the point (-1,5). Let's find the other one.
We have another point (1,1). We can see these are the points we graphed on the line. We have found them both geometrically and algebraically.
Now we can look at the inequality sign. We notice that we have a less than or equal to sign (≤).
Let's first think about the equal to part of the inequality. When graphing inequalities in one variable, we would draw circle around the value and shade the circle because it is included in the inequality. With equations in two variables, we don't have a point - we have a line. We would treat the line in a similar fashion, by bolding the line to denote that every point on the line is included in the inequality. In other words, every x and y value on the line will make the inequality statement true.
Now we have to think about the less than part of the inequality. For inequalities of one variable, we would shade a line. In this case, we will need to shade a region on one side of the line. Intuitively, if we wanted to shade a region less than the line, we would shade the region to the left, or under the line. Most of the time, the region below the line will be less than and the region above the line will be greater than. Let's shade the region below the curve.
To make sure this is the correct region, we can pick any point in the region, plug it into the inequality and see if the statment is true. We can see that the region contains the origin, so let's plug in the point (0,0).
This is a true statement, so we have graphed the correct region for the inequality.
We can also make a table of points located inside and outside of the region and see if they satisfy the linear inequality when substituted for x and y.
Let's try another example
(2) Graph the linear inequality
We can see that this linear inequality is in Standard Form (Ax + By = C). We could easily find the x and y intercepts by setting each variable to 0 and solving for the other, but let's put it in slope-intercept form.
Remember that when we divide by a negative number, we need to flip the inequality sign. Now we have a less than instead of a greater than sign. Now that we have the inequality in slope intercept form (y = mx+b), let's graph the line and worry about the inequality later. The y intercept is b = -2, so the point is (0,-2). Our slope is m = 1/2, wo we can rise 1 and run 2 to find the next point.
Now that we have our line, we can look at the inequality sign.
We notice that we now have a less than sign (<). In one variable inequalities, we would put an open circle around the value to denote that it is not contained in the inequality. Similarly, since it is only less than and not less than or equal to, we will need to make the line a dotted line to denote that every value on the line is not contained in the inequality.
Since it is less than, we will graph the region under the line. To check, we can either use a point in the region and see if the statement is true, or pick a point not in the region and see if the statement is false. It is easy to use the origin (0,0) because the number 0 is easy to operate on. Since it is not in the region we want to check, we will plug it in the inequality and see if the statement is false.
This statement is indeed false, so the region containing the origin is not the shaded region.