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Graphing Linear Inequalities

Author: Sophia

what's covered
In this lesson, you will learn how to identify the corresponding graph of a linear inequality. As you'll discover, graphing linear equalities builds your results driven skill. Specifically, this lesson will cover:

Table of Contents

1. Graphing a Two-Variable Inequality on the Coordinate Plane

When graphing a linear inequality, we generally follow these steps:

step by step
1. Analyze the inequality symbol to determine a solid or dashed line.

First, we need to decide if the graph of the linear inequality will have a solid or a dashed line. Solid lines indicate that the exact value of y is included in the solution to the inequality. We use solid lines when the inequality symbol is non-strict, or includes "or equal to." These would be our inequality symbols ≤ and ≥.

On the other hand, we use dashed lines when the exact value of y is not allowed in our solution to the inequality. Dashed lines represent strict inequalities, which do not include "or equal to." These are the symbols < and >.

Endpoint IS included in interval Endpoint IS NOT included in interval
less or equal than or greater or equal than < or >
A solid line A dashed line

Use solid lines when graphing inequalities with the non-strict symbols ≤ and ≥. Use dashed lines when graphing inequalities with strict symbols < and >.

2. Graph the equation of the line as y equals using either a solid or dashed line.

Once we have determined what kind of line to use when graphing the inequality, we need to actually graph it. To do so, we somewhat ignore the fact that we are dealing with an inequality, and simply graph the line as if it were y equals rather than with an inequality symbol. Just remember to use the appropriate line.

3. Determine a solution region to shade either above or below the line.

We're not finished with the graph of our inequality yet. Next, we need to shade in half of the coordinate plane to represent the solution region. The solution region highlights all possible x- and y-coordinates that satisfy the inequality. When boundary lines are in the form y equals (with the variable y isolated on one side), it is simple enough to once again examine the inequality symbol to see which portion of the coordinate plane to shade.
  • If the inequality symbol is "greater than y" or "greater than or equal to y", we shade everything above the line.
  • If the inequality symbol is "less than y" or "less than or equal to y", we shade everything below the line.

EXAMPLE

Graph the linear inequality y less than 2 x minus 5.

Step Explanation Graph
1. Use solid or dashed line. To start, we note that the inequality symbol is strict, thus we will use a dashed line. A dashed line
2. Graph the equation of the line as y equals. Next, we note the y-intercept of -5 and the positive slope of 2 to create the start of our graph. A graph with an x-axis and a y-axis, both ranging from −6 to 6. A dashed line slants upward from left to right, passing through the y-intercept at −5 and covering the third, fourth, and first quadrants.
3. Shade above or below the line. Since our inequality is y less than 2 x minus 5, a less-than inequality, we know to shade everything below our dashed line. A graph with an x-axis and a y-axis, both ranging from −6 to 6. A dashed line slants upward from left to right, passing through the y-intercept at −5 and covering the third, fourth, and first quadrants. The area below the dashed line is shaded.

This is the graph of our inequality. The dashed line tells us that the exact values of x and y that fall on the line are not included in our solution. The line y equals 2 x minus 5 is considered the boundary line, where coordinates on one side of the line are solutions, and coordinates on the other side of the line are non-solutions. The highlighted region is known as the solution region, as it shows all possible x- and y-values that satisfy the inequality.

Results Driven: Skill in Action
Let's say that you sell solar panels and you are setting a sales goal for the next 12 months. Having knowledge of graphs and linear equations, as well as your sales numbers for the past five years, you decide to make a graph in order to determine what your sales goal should be. You use the x-axis to show the number of months and the y-axis to show the number of solar panels. You don't want to overestimate your sales, so you decide to calculate the minimum number of solar panels you estimate that you will sell. In other words, after x months, you will sell y number of solar panels or greater. By determining the equation for the linear inequality and then graphing this equation, you can determine your goal, strengthening your results driven skill.


2. Using a Test Point to Shade the Solution Region

While using the above method to determine which half of the coordinate plane to shade, there is a more mathematically sound method. This method will be particularly useful when graphing non-linear inequalities, where "above the line" and "below the line" aren't as clear cut.

A test point is any coordinate (x, y) that does not lie on the boundary line. We can think of it as being a representative for all other coordinates on that side of the boundary line. If our test point satisfies the inequality, it represents a solution, as do all the other points on that side. If our test point does not satisfy the inequality, then it represents a non-solution, and the other side of the boundary line should be shaded.

EXAMPLE

Consider the example from above. Suppose we have completed Step 2 and have drawn our boundary line. Now we need to determine which side to shade.

A graph with an x-axis and a y-axis, both ranging from −6 to 6. A dashed line slants upward from left to right, passing through the y-intercept at −5 and covering the third, fourth, and first quadrants.

Let's use the test point (0, 0). This is a good strategy, so long as the origin is not on the graph itself.

y less than 2 x minus 5 Our inequality
0 less than 2 open parentheses 0 close parentheses minus 5 Use (0, 0) as our test point.
0 less than 0 minus 5 Evaluate 2(0).
0 less than short dash 5 Evaluate 0 minus 5. 0 less than short dash 5 is a false statement, meaning (0, 0) is not in solution region

We determined that the origin is not part of the solution region, because the point (0, 0) does not satisfy our inequality. This means that all points on the other side of the boundary line are solutions, and we should shade in that half-plane on the graph, as shown below.

A graph with an x-axis and a y-axis, both ranging from −6 to 6. A dashed line slants upward from left to right, passing through the y-intercept at −5 and covering the third, fourth, and first quadrants. The area below the dashed line is shaded.

hint
Choosing (0, 0) whenever possible is wise, because it is easy to multiply by 0 and add 0, making our calculations easier.

big idea
A test point is a coordinate point (x, y) that is a representative for all points on one side of the boundary line. Use the coordinates to plug into x and y in the inequality statement. If the statement is true, it represents a solution, as does that entire half-plane. If the statement is false, it represents a non-solution, and the other half-plane is the solution region.

summary
In this lesson, you learned that when graphing a two-variable inequality on the coordinate plane, the symbols "less than" and "greater than" indicate using a dashed line, and the symbols "less than or equal to" and "greater than or equal to" indicate using a solid line. When determining which region to shade on the graph, the symbols "less than" and "less than or equal to" indicate shading below the line. The symbols "greater than" and "greater than or equal to" indicate shading above the line. You also learned that using a test point to shade the solution region can be helpful in determining the correct region. The x- and y-coordinates of a point that is in the shaded region will yield a true statement when substituted into the inequality. The x- and y- coordinates of a point that is not in the shaded region will yield a false statement when substituted into the inequality.

Best of luck in your learning!

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