Algebra – Solving Systems of Linear Inequalities Objective: Students will solve systems of linear inequalities by graphing.

Before we begin… In previous lessons we have explored different ways to solve systems of linear equations… In this lesson we will look at linear inequalities … Essentially, you will graph the linear system of inequalities on the same coordinate plane, shade the solution area for each inequality. The portion of the coordinate plane where the shading overlaps represents the solution to the system of linear inequalities.

Review We have already worked with some of this material…as a quick review, you should already know that when graphing inequalities: are represented as a dashed line ≤ and ≥ are represented as a solid line The shaded portion of the coordinate plane represents the solution set to the inequality. That is, any point in the shaded area, when substituted, will make the inequality true

Comments I cannot stress the importance of being organized and laying out your work here… The same strategies you used to graph equations will be used to graph inequalities… It is not enough to be able to mechanically graph the inequalities…you are also expected to be able to interpret the results… That is, you must be able to read the graph and determine where and what the solution set is… The key here is to analyze the inequalities first!

Process The process for solving systems of linear inequalities is: Step 1 – Write the inequality in a format that is easy to graph Step 2 – Graph and shade the solution set for each of the inequalities on the same coordinate plane Step 3 – Identify the area where the shading overlaps Step 4 – Choose a point in the overlapping shaded area and substitute it into each of the inequalities and determine if you get a true or false statement.

Example #1 Solve the system of linear inequalities by graphing. y < 2Inequality #1 x ≥ -1Inequality #2

Example #1 y < 2Inequality #1 x ≥ -1Inequality #2 Step 1 – Write the inequality in a format that is easy to graph The first step is to analyze the inequalities. I see that all the inequalities are in a format that I can easily graph…Therefore, I do not need to do this step. Something to think about…In the back of my mind I see that inequality #1 & #2 have only 1 variable…from working with equations I know that an equation in 1 variable produces either a horizontal or vertical line…the same holds true for inequalities…I already have a picture of what the graph will look like in the back of my mind…

Example #1 Step 2 – Graph and shade the solution set for each of the inequalities on the same coordinate plane y x y < 2Inequality #1 x ≥ -1Inequality #2 y < 2 x ≥ -1

Example #1 y x y < 2Inequality #1 x ≥ -1Inequality #2 y < 2 x ≥ -1 In this example the square where the 2 solution sets overlap represents the solution set to the system of inequalities Step 3 – Identify the area where the shading overlaps

Example #1 Step 4 – Choose a point in the overlapping shaded area and substitute it into each of the inequalities and determine if you get a true or false statement y < 2Inequality #1 x ≥ -1Inequality #2 In this example the origin (0, 0) lies within the solution set. I will use that point to determine if the solution set is correct by substituting the values of x and y into the original inequalities y < 2 0 < 2 True x ≥ -1 0 ≥ -1 True

Comments When choosing a point in the overlapping shaded area be careful if you choose a point on the line… If the line is dashed ( ) the points on the line are not included in the solution set If the line is solid ( ≤ or ≥) the points on the line are included in the solution set.

Example # 2 Sometimes you are given a graph of a system of linear inequalities and are asked to write the system of inequalities. Again, it is expected that you can read the graph and determine the inequalities that the graph represents… Let’s look at an example…

Example # 2 Write a system of linear inequalities that defines the shaded region to the right y x Line #1 Line #2

Example #2 y x Line #1 Therefore, the inequality for line #1 is written as y < 3 In this example I see that Line #1 crosses the y-axis at +3. I see that a dashed line is used so I will use the symbol I also see that the area below the line is shaded. That means the value is less than. Let’s look at Line #1 first

Example #2 y x Line #2 Therefore, the inequality for line #2 is written as y > 1 I see that Line #2 crosses the y-axis at +1. Again, I see that a dashed line is used so I will use the symbol I also see that the area above the line is shaded. That means the value is greater than. Now let’s look at Line #2

Example #2 y x Line #1 Line #2 After analyzing the graph we can now determine the system of inequalities that the graph represents as: y < 3 y > 1

Comments For some reason students have a hard time with reading graphs… The expectation is if you are given an equation or inequality and you know how to graph it using slope-intercept form…then you should be able to look at a graph, pick out the parts of the slope-intercept form and determine the equation or inequality of the graph…

Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn Write a system of linear inequalities that define the shaded regions y x y x y x

Your Turn Graph the system of linear inequalities. 4. 2x + y > 2and6x + 3y < x – 2y < 6andx – y < 9 6. x – 3y ≥ 12andx – 6y ≤ x + y ≤ 6andx ≥ 1andy ≥ 0

Your Turn Graph the system of linear inequalities 8. 3/2x + y < 3 x > 0 9. y ≥ 0 y ≤ /2x + y ≤ 3 4x + y < 2

Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about systems of linear inequalities. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…