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P-3 Linear Equations and Inequalities Vocabulary Linear Equation in one variable. Ax + B = C A≠0 B and C are constants You’ve seen this before! 4x – 2 = 6 + 2 +2 4x = 8 ÷ 4 ÷4 x = 2 Give me another example of a linear equation in one variable. Linear Equation in 2 Variables Ax + By = C 3x + 4y = 12 Linear Equation in 3 Variables 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. 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 Solutions of Linear equations What does “solution” mean ? Vocabulary Solution: the number the variable must equal in order to make the statement true. x+1=2 The solution is: x = 1 “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 order to make the statement true. x+1=2 -1 -1 x = 1 (subtraction property of equality) (solution) 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 5 - (x - 4) =5–x–8 NO NO NO!!! NO NO NO!!! Your turn: Solve these equations. 3. 3 x 2 1. 4 3 2. 5x 1 3 7 4. 2x 7 3 4x 2x 3 2 3x 5 5 Vocabulary Linear Inequality (in one variable): Ax + B < C or 3x – 2 < 4 Ax + B > C or Ax + B ≤ C or Ax + B ≥ C. More Vocabulary Equivalent Equations: An equation that has the same solution as the original equation. x+2=4 -2 -2 x=2 (subtraction property of equality) Equivalent Inequalities: An inequality that has the same solution as the original inequality. x+2<4 -2 -2 x<2 (subtraction property of inequality) Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 -2x -2x x+1 ≤ 6 -1 -1 x≤5 KEY POINT: collect variable on the side that will result in a positive coefficient. Your turn: Solve the inequality ≤ 6 5. 2x + 2 6. 2(x – 3 ) ≥ 7. -14x – 2 8 < 5x + 6 The “Gotcha” of Inequalities 2 – 2x + 2x ≤ 6 2 – 2x + 2x -2 2 ≤ 2x + 6 -6 -6 -4 ≤ 2x ÷2 ÷2 -2 ≤ x ≤ 6 -2 -2x ≤ ÷ -2 x 4 ÷ -2 ≤ -2 Anytime you multiply or divide by a negative number, you must switch the direction of the inequality !! Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 -2x -2x x+1 ≤ 6 -1 -1 x≤5 To avoid the “gotcha”: collect the variable on the side that will result in a positive coefficient. Your turn: Solve the inequality 8. 2x – 6 ≤ 3 – x 9. 18 + 2x ≥ 9x + 4 10. 2(x – 4) < 4x + 6 Compound inequalities (two inequality symbols) 5 ≤ x+1< 9 5 ≤ x+1 -1 4 ≤ x Same as: and x+1< 9 -1 -1 and 4 ≤ x < 8 -1 x < 8 Compound inequalities (two inequality symbols) 5 ≤ x+1< 9 -1 4 ≤ x -1 -1 KEY POINT: subtraction property of inequality do the same thing (left-middle-right) < 8 Same as: 4 ≤ x and x < 8 Your turn: Solve the inequality 11. -3 < 4 – x ≤ 3 12. -5 < x + 1 and x + 1 ≤ 6 Solving inequalities (“or” type) KEY POINT: treat “or” type compound inequalities as two separate inequalities. x-2 ≤ 3 +2 +2 or x ≤ 5 or x ≤ 5 or x+2> 8 -2 -2 x > x > 6 6 Your turn: Solve the inequality 13. 14. 4x - 7 ≤ 5 or 3x + 2 > 23 x + 1 ≤ -3 or x – 2 > 0 Sometimes there is no solution Solution: the value(s) of the variable that make the statement true. 2(x – 4) > 2x + 1 2x – 8 -2x –8 > 2x + 1 -2x > 1 No solution: when the variable dissappears and the resulting statement is false. Sometimes the solution is all real numbers. Solution: the value(s) of the variable that make the statement true. 4x – 5 4x – 5 -4x ≤ 4(x + 2) ≤ 4x + 8 -4x –5 ≤ 8 Infinitely many solutions: when the variable dissappears and the resulting statement is true. Your turn: Solve the inequality 15. 2(3x – 1) > 3(2x + 3) 16. 2x + 3 ≤ 3(x + 2) – x Graphing Single Variable inequalities x>3 What part of the number line is greater than 3 ? 1 2 3 4 5 Graphing Single Variable inequalities x<5 What part of the number line is less than 5 ? 1 2 3 4 5 Your turn: Graph the following 17. x≥7 18. 3>x Graphing Compound Inequalities x>3 and x<5 And means both conditions must be met What part is x>3? What part is x < 5? What is the intersection or overlap of the two? 1 2 3 4 5 Vocabulary Compound inequality x>3 and Hint: This can also be written as: 3<x<5 Hint: Inequality with “and” looks like: 1 2 3 4 5 x<5 Your turn: 19. 20. x>2 Graph the following compound inequalities. and -2 < x ≤ 5 x<6 Graphing “or” type compound inequalities. x≤3 or x>5 Or means: the points that satisfy either condition Which part is x ≤ 3 ? 1 2 3 Which part is x > 5 ? 4 5 Hint: inequality with “OR” looks like: Your turn: 21. Solve and graph the compound inequality: 2x + 3 ≤ 5 or x - 3 > 2 1 2 3 4 5 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. Your turn: 22. 23. (a) Write in inequality notation (b) Graph the inequality There are least 65,000 spectators at the game. It never gets above 100 degrees in Huntsville. 24. You can fit, at most, 5 cars in your garage. Three Ways to show an Inequality x≤2 x>3 1. Inequality: 2. Bracket Notation: (3, ) 3. Number line Notation: 1 2 3 4 5 6 ( , 2] Inequalities Involving Fractions x 1 x 1 3 2 4 3 Another example? x5 x2 1 8 2 3 Your Turn: Solve this inequality 2 x 3 3x 1 x 1 25. 2 5 3 Homework P-3: evens: 2-10, 18-26, 32-44, 54 (18 problems)