Linear Inequalities in Two Variables Lesson

Now that we have completely mastered graphing linear equations in two variables, we move into graphing a linear inequality in two variables. A linear inequality in two variables is of the form:
ax + by < c
where a, b, and c are any real numbers, a and b are not both zero, and the symbol "<" can be ">", "≤", or "≥". When we deal with the solution set for an inequality, we are normally dealing with a range of values. This means any point (x,y) that makes the inequality true is part of our solution set. When we graph a linear inequality in two variables, we will see three distinct parts:

• The Solution Region
  • This region contains all ordered pairs (x,y) that satisfy the inequality
• The Non-Solution Region
  • This region contains all ordered pairs (x,y) that do not satisfy the inequality
• The Boundary Line
  • The boundary line separates the solution region from the non-solution region
  • The boundary line is drawn as a dashed line for a strict inequality: "<" or ">"
  • The boundary line is drawn as a solid line for a non-strict inequality: "≤" or "≥"

There are two methods we can use to graph a linear inequality in two variables. The first and more tedious method involves the use of a test point.

Graphing a Linear Inequality in Two Variables using the Test Point Method

• Graph the boundary line
  • We graph the boundary line by replacing our inequality symbol with an equality symbol. We then graph the equation. This line is drawn as a dashed line for a strict inequality and a solid line for a non-strict inequality.
• Choose a test point (x,y)
  • Pick any point on the coordinate plane that is not on the boundary line. From this, we can plug into our original inequality. If the test point works as a solution, we shade the side of the boundary line which contains the test point. If the test point does not work as a solution, we shade the opposite side of the boundary line (the side of the boundary line which does not contain the test point).

Before we jump into an example, let's make a few things clear. On the coordinate plane, we can only be in one of three places: the solution region, the non-solution region, or on the boundary line. Based on the inequality symbol, we know if the boundary line is part of the solution set. When we have a strict inequality "<" or ">", the boundary line is not part of the solution set. To show this, we draw our boundary line as a dashed line. Points on the line will not satisfy the inequality. On the other hand, when we have a non-strict inequality "≤" or "≥", points on the boundary line are part of the solution set. To show this, we draw the boundary line as a solid line. Points on the boundary line will satisfy the inequality. Additionally, we know that if we are not on the boundary line we will be in the solution region or the non-solution region. If we pick any point that is not on the boundary line, plug into the inequality, and get a true statement, our point lies in the solution region. Otherwise, we are in the non-solution region. Let's look at an example.
Example 1: Graph each.
7x - 2y < -4
Step 1) Let's graph our boundary line
To graph our boundary line, we replace our inequality symbol "<" with the equality symbol "=".
7x - 2y = -4
Since we have a strict inequality, the boundary line will be a dashed line. Step 2) Pick a test point
Since (0,0) is easy to work with, we will choose this as our test point.
7(0) - 2(0) < -4
0 < -4 (false)
Since our test point did not satisfy the inequality, it must lie in the non-solution region. Therefore, we will shade the opposite side of the boundary line or the region of the coordinate plane that does not contain the point (0,0). A quicker method involves solving the inequality for y and then shading based on the inequality symbol. With this method, we will shade above the line for a greater than or greater than or equal to and below the line for a less than or a less than or equal to. Note, this only works when the inequality is solved for y. If we look at our example above: $$7x - 2y < -4$$ $$-2y < -7x - 4$$ $$y > \frac{7}{2}x + 2$$ Since we have solved for y, we can observe the greater than symbol. This means we want to shade above the line, which is what we found using the test point method. Let's look at another example.
Example 2: Graph each.
5x - y ≤ 5
Let's use the quicker method. We will solve for y first:
y ≥ 5x - 5
We want to graph a solid boundary line of: y = 5x - 5, and then shade above the line:

Graphing the Union of Two Linear Inequalities in Two Variables

We previously learned that we solve a compound inequality with "or" by finding the union of the two solution sets. Let's look at an example.
Example 3: Graph each compound inequality.
2x - 5y ≤ 5
or
3x - 2y < -4
Let's begin by graphing the first inequality.
2x - 5y ≤ 5 Now we can graph our second inequality.
3x - 2y < -4 The solution for a compound inequality with "or" is found as the union of the two solution sets. This means our solution will include any region of the coordinate plane that satisfies either inequality. From our graph above, we can see three different shaded areas. The part that is shaded yellow satisfies only 3x - 2y < -4, the part shaded in pink satisfies only 2x - 5y ≤ 5. Lastly, the part that is shaded in orange satisfies both inequalities. Since our inequality uses the word "or", all of these shaded areas are included in the solution set.

Graphing the Intersection of Two Linear Inequalities in Two Variables

We also learned how to solve a compound inequality with "and". When we solve a compound inequality with "and", we want to find the intersection of the two solution sets. This means we want to find the region of the coordinate plane that satisfies both inequalities. Let's look at an example.
Example 4: Graph each compound inequality.
4x - y ≤ 0
and
x - 5y > -20
Let's begin by graphing the first inequality.
4x - y ≤ 0 Now we can graph our second inequality.
x - 5y > -20 The solution for a compound inequality with "and" is found as the intersection of the two solution sets. This means our solution will include any region of the coordinate plane that satisfies both inequalities.

---

Skills Check: