1. P-3 Linear Equations and Inequalities

3. Vocabulary Linear Equation in one variable. Ax + B = C A ≠ 0 B and C are constants You’ve seen this before! 4x – 2 = 6 Give me another example of a linear equation in one variable. + 2 +2 4x = 8 ÷ 4 ÷4 x = 2 x = 2

4. Linear Equation in 2 Variables Ax + By = C Linear Equation in 3 Variables 3x + 4y = 12 Ax + By + Cz = D 3x + 4y + 6z = 12 There is no limit to the number of variables in a linear equation. Animation uses over 100.

5. What makes it a ‘linear” equation ? If the exponent of the variable(s) is a ‘1’, then it is a linear equation. 3x + 4y + 6z = 12

6. Solutions of Linear equations What does “solution” mean ? Vocabulary Solution: the number the variable must equal in order to make the statement true. in order to make the statement true. x + 1 = 2 x + 1 = 2 The solution is: x = 1 The solution is: x = 1

7. “Solving” Linear equations What does it mean to “solve” an equation? Vocabulary Solve: using properties to re-write the equation in the form: x = (some exact value) in the form: x = (some exact value) in order to make the statement true. in order to make the statement true. x + 1 = 2 x + 1 = 2 (subtraction property of equality) (subtraction property of equality) -1 -1 -1 -1 x = 1 x = 1 (solution) (solution)

8. Distributive Property of Addition over Multiplication 2(x + 4)= (2 * x)+(2 * 4)= 2x + 8 Biggest 2 errors in the distributive property: Trying to multiply when the operation is add or subtract Failing to distribute a negative to BOTH terms inside parentheses 5 - (x - 4)= 5 x – 20 NO NO NO!!! = 5 – x – 8 5 - (x - 4) NO NO NO!!!

9. Your turn: Solve these equations. 1. 2. 3. 4.

10. Vocabulary Linear Inequality (in one variable): Ax + B < C or Ax + B > C or Ax + B ≤ C or Ax + B ≥ C. 3x – 2 < 4

11. More Vocabulary Equivalent Inequalities:. Equivalent Inequalities: An inequality that has the same solution as the original inequality. x + 2 = 4 x = 2 x + 2 < 4 -2-2 -2-2 x < 2 (subtraction property of inequality) (subtraction property of equality) Equivalent Equations: Equivalent Equations: An equation that has the same solution as the original equation.

12. Solving inequalities Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 KEY POINT: collect variable on the side that will result in a positive coefficient. -2x x + 1 ≤ 6 x ≤ 5 x ≤ 5

13. Your turn: Solve the inequality 7. 7. -14x – 2 < 5x + 6 5. 5. 2x + 2 ≤ 6 6. 6. 2(x – 3 ) ≥ 8

14. The “Gotcha” of Inequalities 2 – 2x ≤ 6 + 2x 2 ≤ 2x + 6 -6 -4 ≤ 2x ÷2 -2 ≤ x 2 – 2x ≤ 6 -2 -2x ≤ 4 ÷ -2 x ≤ -2 Anytime you multiply or divide by a negative number, you must switch the direction of the inequality !!

15. Solving inequalities Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 To avoid the “gotcha”: collect the variable on the side that will result in a positive coefficient. -2x x + 1 ≤ 6 x ≤ 5 x ≤ 5

16. Your turn: Solve the inequality 10. 2(x – 4) < 4x + 6 8. 2x – 6 ≤ 3 – x 9. 18 + 2x ≥ 9x + 4

17. Compound inequalities Compound inequalities (two inequality symbols) 5 ≤ x + 1 and 4 ≤ x 4 ≤ x Same as: 4 ≤ x < 8 x + 1 < 9 and x < 8 x < 8 5 ≤ x + 1 < 9

18. Compound inequalities Compound inequalities (two inequality symbols) 5 ≤ x + 1 < 9 KEY POINT: subtraction property of inequality  do the same thing (left-middle-right) -1 -1 -1 4 ≤ x < 8 4 ≤ x < 8 4 ≤ x and x < 8 Same as: 4 ≤ x and x < 8

19. Your turn: Solve the inequality 11. 11. -3 < 4 – x ≤ 3 12. 12. -5 < x + 1 and x + 1 ≤ 6

20. Solving inequalities Solving inequalities (“or” type) x - 2 ≤ 3 or x + 2 > 8 KEY POINT: treat “or” type compound inequalities as two separate inequalities. +2 x ≤ 5 x ≤ 5 -2 -2 -2 -2 x > 6 x > 6or x ≤ 5 x ≤ 5 x > 6 x > 6or

21. Your turn: Solve the inequality 13. 13. 4x - 7 ≤ 5 or 3x + 2 > 23 14. 14. x + 1 ≤ -3 or x – 2 > 0

22. Sometimes there is no solution 2(x – 4) > 2x + 1 Solution: the value(s) of the variable that make the statement true. the statement true. 2x – 8 > 2x + 1 -2x -2x – 8 > 1 – 8 > 1 No solution: when the variable dissappears and the resulting statement is false.

23. Sometimes the solution is all real numbers. Solution: the value(s) of the variable that make the statement true. 4x – 5 ≤ 4(x + 2) -4x – 5 ≤ 8 Infinitely many solutions: Infinitely many solutions: when the variable dissappears and the resulting statement is true. 4x – 5 ≤ 4x + 8

24. Your turn: Solve the inequality 15. 15. 2(3x – 1) > 3(2x + 3) 16. 16. 2x + 3 ≤ 3(x + 2) – x

25. Graphing Single Variable inequalities x > 3 1 2 3 4 5 What part of the number line is greater than 3 ?

26. Graphing Single Variable inequalities x < 5 1 2 3 4 5 What part of the number line is less than 5 ?

27. Your turn: Graph the following 17 17. x ≥ 7 18. 18. 3 > x

28. Graphing Compound Inequalities x > 3 and x < 5 1 2 3 4 5 What part is x > 3 ? And means both conditions must be met What part is x < 5? What is the intersection or overlap of the two?

29. Vocabulary x > 3 and x < 5 Hint: Inequality with “and” looks like:   Compound inequality 1 2 3 4 5 Hint: This can also be written as: 3 < x < 5

30. Your turn: Graph the following compound inequalities. x > 2 and x < 6 19. 20. -2 < x ≤ 5

31. Graphing “or” type compound inequalities. x ≤ 3 or x > 5 Or means: the points that satisfy either condition 1 2 3 4 5 Which part is x > 5 ?Which part is x ≤ 3 ? Hint: inequality with “OR” looks like:  

32. Your turn: 21: 21. Solve and graph the compound inequality: 1 2 3 4 5 2x + 3 ≤ 5 or x - 3 > 2

33. Verbal Inequalities The cost of a car is at most $20,000. It takes Jehah no less than 5 minutes to run a mile. It takes between 3 and 8 months to build a house. The cost of a loaf of bread is less than $2 You can’t buy a car for less than $8000.

34. Your turn: Your turn: (a) Write in inequality notation (b) Graph the inequality It never gets above 100 degrees in Huntsville. 22.There are least 65,000 spectators at the game. 23. 24. You can fit, at most, 5 cars in your garage.

35. Three Ways to show an Inequality 1. Inequality: x > 3 2. Bracket Notation: (3, ) 3. Number line Notation: x ≤ 2 1 2 3 4 5 6 (, 2]

36. Inequalities Involving Fractions

37. Another example?

38. Your Turn: Solve this inequality 25.

39. Homework P-3: evens: 2-10, 18-26, 32-44, 54 (18 problems)