Ch 6.6 Day 1 – Solving Systems of Linear Inequalities Algebra 1 Ch 6.6 Day 1 – 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: < and > 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 < 2 Inequality #1 x ≥ -1 Inequality #2 y > x – 2 Inequality #3

Example #1 y < 2 Inequality #1 Step 1 – Write the inequality in a format that is easy to graph x ≥ -1 Inequality #2 y > x – 2 Inequality #3

Example #1 y x Step 2 – Graph and shade the solution set for each of the inequalities on the same coordinate plane y < 2 Inequality #1 x ≥ -1 Inequality #2 y > x – 2 Inequality #3

Example #1 y Step 3 – Identify the area where the shading overlaps y < 2 Inequality #1 x ≥ -1 Inequality #2 y > x – 2 Inequality #3

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 < 2 Inequality #1 x ≥ -1 Inequality #2 y > x – 2 Inequality #3 y > x – 2 x ≥ -1 y < 2

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 ( < or >) 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.

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

Your Turn Solutions #1 - 4 to check your work, choose a point in the solution set and substitute it into the original inequalities. If you get a true statement than you graphed the inequalities correctly. If not you did something wrong…go back and do some error analysis…If you cannot find your error bring your work to me and we will look at it together…