1. 10.5 Systems of Linear Inequalities and Applications 11.1 Interval Notation and Linear Inequalities

2. Process and tips When we graph systems of linear inequalities we will be graphing 2 inequalities in the xy-plane. This is exactly what we did in 9.6 but doing it two times. We will now have 2 places where we have shaded. The overlap of the two is the solution to the system. Be careful when you want to use the idea of shading above and below you want to make sure that the equation is in y = mx+b form ( = sign will be a ). In standard form you cannot make the distinction correctly (this is the mistake I made on the first quiz) However if it is in standard form it is probably less work to choose (0,0) and use that point as a test point to help make the decision of where to shade.

7. A college sports arena plans on charging $10 for certain seats and $6 for others. They want to bring in more than $12,000 from all ticket sales and have reserved at least 300 tickets for the $6 rate. Determine a system of inequalities describing all possible ticket combinations that would provide the desired results. Define Variables: let x=10$ tickets and y=6$ tickets We know that we cannot sell negative ticket so that means x≥0 and y ≥0. (sometimes called non-negative constraints). We also know that we have 300 $6 reserved seats sold. y ≥ 300. Now we have to create a revenue inequality. We do not know how many tickets we are selling total and we do not know how many of each type of seat we are selling but we do know the cost of each seat and we know a desired level of revenue so 10x + 6y > 12,000

8. Now we can use all of these inequalities together on a graph to determine the correct solution. Because we have to satisfy all the inequalities at one time if y ≥ 300 and y ≥0 then we do not need to graph the y ≥ 0 inequality Its so much easier to graph these if you use intercepts

11. Graph the solution set

12. 11.1 Interval Notation and Linear Inequalities

13. A linear inequality is an inequality that can be put in the form ax+b < c, where a, b, and c are real numbers. Simply stated inequalities are statements that state one side of the, or ≤, ≥ is different than (or unequal) from the other. When you solve linear inequalities you approach them just like you approach equations and try to isolate the single variable. – The only difference occurs when you multiply/divide by a negative value. – Also your solution is a set of infinite values instead of just 1 solution.

14. There are three different ways to express your solution to this inequality. Graphical Set Notation Interval Notation (read like a number line, L to R)

15. These symbols, whether on a graph or in interval notation indicate that the actual number is not included as an answer. These symbols, whether on a graph or in an interval notation indicate that the actual number IS included as an answer.

16. Graph the following inequalities, provide the solution in set notation and interval notation. InequalityGraphInterval Notation Set Notation

17. Examples

18. Example