Unit 4 - Equations and Inequalities - Vocabulary

Transcription

1 12/5/17 Unit 4 Unit 4 - Equations and Inequalities - Vocabulary Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

2 Review Vocabulary: Absolute value Additive inverse Coefficient Distributive Property Numerical Expression Algebraic Expression Integer Opposite Rational number Reciprocal

3 Equation A mathematical sentence that includes an equals sign to compare two expressions. Example: 2x + 1 = 9

4 Substitution To replace variables in an expression with numbers.

5 Evaluate Substitute a value for each variable, and then follow the order of operations. Example: evaluate: 2x + 1, when x = = 7

6 Solve an Equation Apply inverse operations in order to isolate the variable. Example: x + 2 = 8 x = 8 2 x = 6

7 Complex Fraction A fraction where the numerator, denominator, or both contain a fraction. Example: Main Fraction Bar

8 Solve a Complex Fraction Rewrite the complex fraction as a division expression. The main fraction bar becomes a division sign = = = 15 4

9 12/6/17 Unit 4 Unit 4 Solving One-Step Equations Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

10 The Golden Rules of Equation Solving 1. Undo what has been done to the variable. 2. Always do the same thing to both sides of the equation.

11 To solve an equation you perform inverse operations. Operation Inverse Operation + +

12 Al Karaba Solving Equations 1 Addition & Subtraction (see video assigned in digits)

13 Example 1 x + 12 = 4 x = 4 12 x = 8

14 Example 2 7 = x = x = x

15 Al Karaba Solving Equations 2 Multiplication & Division (see video assigned in digits)

16 Example 3 5x = 35 5x 5 = 35 5 x = 7

17 Example 4 4 = x 3 4(3) = x 3 (3) 12 = x

18 12/11/17 Unit 4 Solving Two-Step Equations Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

19 To simplify expressions use proper order of operations (GEMDAS) Multiplication occurs before addition

20 To solve equations use inverse operations. Because we are undoing operations, this means doing GEMDAS backwards.

21 SADMEG I don t know how to solve two-step equations!

22 Everything You Need to Know About Solving Two-Step Equations To solve equations we undo addition and subtraction before we undo multiplication and division.

23 Al Karaba Solving Two-Step Equations Solving Two-Step Equations (see video assigned in digits)

24 Example 1: 5x + 7 = x = Undo addition before undoing multiplication x = 5

25 Example 2: Undo subtraction x 3 8 = 5 before undoing division x 3 = x = 9

26 Example 3: 2x + 5 = x = x = 14 Undo addition before undoing multiplication 2 2 x = 7

27 12/12/17 Unit 4 Begin on a new page Solving Equations (if the variable is being multiplied by a fraction) Write the date and unit in the top corners of the page Write the title across the top line

28 To solve an equation in which the variable is being multiplied by a fraction: Multiply both sides of the equation by the reciprocal of the fraction.

29 Example 1: x = x = 60 2 = 30

30 Example 2: x = x = 88 4 = 22

31 Example 3: 3 5 x 1 = x = x = 15 Undo subtraction before undoing multiplication by a fraction

32 1/2/18 Unit 4 Solving Equations with the Distributive Property Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

33 How can I solve the following? 3 4 8x 4 = 1 Method 1: Use the Distributive Property Method 2: Use SADMEG

34 Example 1: Dist. Prop x 4 = x 3 4 (4) = 1 6x 3 = x = x = 2 3

35 Example 1: SADMEG x 4 = Operations on x: or x 4 = x = x = x = 2 3

36 Example 2: Dist. Prop x 5 = x 1 2 (5) = x = x = 21 2 x = 7 2 3

37 Example 2: SADMEG x 5 = Operations on x: or x 5 = x = x = 7

38 1/4/18 Unit 4 Solving Inequalities Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

39 h 48

40 d 70%

41 a 12

42 2x 5 > 6

43 Inequality A mathematical sentence that uses <, >,,, or to compare the value of two expressions. Example: x < 8

44 Solution of an Inequality Any value of the variable that makes the inequality true. x < 8 x = 7.99, 6, 1 3, 567

45 Solve an Inequality Apply inverse operations in order to isolate the variable. Example: x 5 < 11 x < x < 16

46 Symbol Meaning < Less than Less than or equal to > Greater than Greater than or equal to Not equal to

47 Graphing Inequalities Inequality < > or or or Graphing Symbol Meaning The number graphed is NOT part of the solution The number graphed IS part of the solution

48 Example #1: x 2

49 Example #2: x + 7 < x < 4

50 Example #3 x x 1

51 1/9/18 Unit 4 Solving Inequalities 2 Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

52 Consider: 2 < < < 10

53 Consider: 2 < 5 ( 2) 2 < ( 2) 5 4 > 10

54 Consider: 3 < 9 3Τ 3 < Τ 1 < 3 9 3

55 Consider: 3 < 9 3Τ 3 < 9 Τ 3 1 > 3

56 When multiplying or dividing BOTH sides of an inequality by a NEGATIVE number, you must REVERSE the direction of the Inequality symbol.

57 Example #1: 3x < x < 4

58 Example #2: 3x x 4 Reverse the inequality symbol. BOTH sides divided by a negative!

59 1/4/18 Unit 4 Checking the Solution to an Inequality Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line

60 Checking an equation: I solved 4x 11 = 17 and got x = 7. Is it correct? Substitute 7 for x and see if the equation is true: 4x 11 = = = = 17 YES, my answer is correct

61 To check the solution to an inequality: 1. Check the related equation. 2. Check the direction of the Inequality

62 Example 1: Is x < 4 the solution of 4x + 3 < 19? Step 1: Write the inequality as an equation: 4x + 3 = 19 4(4) + 3 = = = 19 YES, the related equation is correct

63 Continued: Is x < 4 the solution of 4x + 3 < 19? Step 2: Pick a value from the solution and see if it works: pick any number less than 4 4x + 3 < 19 4(0) + 3 < 19 3 < 19 YES, solution is correct

64 Example 2: Is x 6 the solution of 3 x 1 5? 2 Step 1: Write the inequality as an equation: x 1 = = = 5 8 = 5 NO, the related equation is incorrect Fix: Start over

65 Example 3: Is x 3 the solution of 5x ? Step 1: Write the inequality as an equation: 5x + 2 = 17 5( 3) + 2 = = = 17 YES, the related equation is correct

66 Continued: Is x 3 the solution of 5x ? Step 2: Pick a value from the solution and see if it works: pick any number less than 3 5x + 2 < 17 5( 5) + 2 < < < 17 NO, inequality is the wrong one! Fix: flip the inequality