1. Linear Inequalities Math 10 – Ms. Albarico

2. Students are expected to: Express and interpret constraints using inequalities. Graph equations and inequalities and analyse graphs, both with and without graphing technology. Investigate and make and test conjectures about the solution to equations and inequalities using graphing technology. Solve problems using graphing technology. Write an inequality to describe its graph. Solve linear and simple radical, exponential, and absolute value equations and linear inequalities. Interpret solutions to equations based on context. Relate sets of numbers to solutions of inequalities.

3. Key Terms Inequality Constraints Feasible Region System of Equation Optimal Solution Objective Function

4. constraint – a restriction on the allowable values of a variable in a problem feasible points – data points that satisfy the constraints given feasible region – a shaded region on a graph indicating that all points within the region are possible solutions to the problem feasible solution – any solution to a problem that is possible within the constraints inequality – a mathematical statement that shows that two numerical or variable expressions are not always equal graphical solutions – solutions obtained by graphing the feasible region. intersection point – the point where two graphs cross each other and where both graphs are equal objective function – a function that allows you to find the maximum or minimum values using given constraints optimal solution – the solution that best meets the constraints in the problem system of equations – two or more equations involving the same variable quantities

5. What ’ s an Inequality? Is a range of values, rather than ONE set number An algebraic relation showing that a quantity is greater than or less than another quantity. Speed limit:

6. Inequality Symbols Less than Greater than Less than or equal to Greater than or equal to Not equal to

7. Solutions…. You can have a range of answers…… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers less than 2 x< 2

8. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers greater than -2 x > -2

9. Solutions continued…. -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers less than or equal to 1

10. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers greater than or equal to -3

11. Did you notice, Some of the dots were solid and some were open? -5 -4 -3 -2 -1 0 1 2 3 4 5 Why do you think that is? If the symbol is > or < then dot is open because it can not be equal. If the symbol is  or  then the dot is solid, because it can be that point too.

12. Write and Graph a Linear Inequality Sue ran a 2-K race in 8 minutes. Write an inequality to describe the average speeds of runners who were faster than Sue. Graph the inequality. Faster average speed > Distance Sue ’ s Time -5 -4 -3 -2 -1 0 1 2 3 4 5

13. Solving an Inequality Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations. Add the same number to EACH side. x – 3 < 5 Solve using addition: +3 x < 8

14. Solving Using Subtraction Subtract the same number from EACH side. -6

15. Using Subtraction… Graph the solution. -5 -4 -3 -2 -1 0 1 2 3 4 5

16. Using Addition… -5 -4 -3 -2 -1 0 1 2 3 4 5 Graph the solution.

17. THE TRAP….. When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

18. Solving using Multiplication Multiply each side by the same positive number. (2)

19. Solving Using Division Divide each side by the same positive number. 33

20. Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. Multiply by (-1). (-1) See the switch

21. Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. -2

22. Linear Inequality Inequality with one variable to the first power. for example: 2x-3<8 A solution is a value of the variable that makes the inequality true. x could equal -3, 0, 1, etc.

23. Transformations for Inequalities Add/subtract the same number on each side of an inequality Multiply/divide by the same positive number on each side of an inequality If you multiply or divide by a negative number, you MUST flip the inequality sign!

24. Ex: Solve the inequality. 2x-3<8 +3 +3 2x<11 2 x< Flip the sign after dividing by the -3!

25. Graphing Linear Inequalities Remember: signs will have an open dot o and signs will have a closed dot graph of 4567-3-20

26. Example: Solve and graph the solution. 6789

27. Compound Inequality An inequality joined by “ and ” or “ or ”. Examples “ and ”“ or ” think between think oars on a boat -4 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 4 5

28. Example: Solve & graph. -9 < t+4 < 10 -13 < t < 6 Think between! - 13 6

29. Last example! Solve & graph. -6x+9 13 -6x 21 x > 1 or x < -7 Flip signs Think oars -71

30. Graphing a Linear Inequality Graphing a linear inequality is very similar to graphing a linear equation.

31. Graphing Inequalities The solution is the set of all points in the region that is common to all the inequalities in that system. It is the region where the shadings overlap. This solution is called the feasible region.

32. Graphing a Linear Inequality 1) Solve the inequality for y (or for x if there is no y). 2) Change the inequality to an equation and graph. 3) If the inequality is, the line is dotted. If the inequality is ≤ or ≥, the line is solid.

33. Graphing a Linear Inequality Graph the inequality 3 - x > 0 First, solve the inequality for x. 3 - x > 0 -x > -3 x < 3

34. Graph: x<3 Graph the line x = 3. Because x < 3 and not x ≤ 3, the line will be dotted. Now shade the side of the line where x < 3 (to the left of the line). 6 4 2 3

35. Graphing a Linear Inequality 4) To check that the shading is correct, pick a point in the area and plug it into the inequality. 5) If the inequality statement is true, the shading is correct. If the inequality statement is false, the shading is incorrect.

36. Graphing a Linear Inequality Pick a point, (1,2), in the shaded area. Substitute into the original inequality 3 – x > 0 3 – 1 > 0 2 > 0 True! The inequality has been graphed correctly. 6 4 2 3

37. More Examples

44. system of equations – two or more equations involving the same variable quantities

46. Practice Exercise

50. Class Activity Answer Investigation 1 & 2 on pages 306- 308, Investigation 3, Part A & B on pages Investigation 4 on pages 319-321

51. Homework CYU # 7, 8, and 9 on pages 308-309. CYU # 12-18 on pages 314-316. CYU # 19-22 on pages 317-318