Top Banner
LINEAR EQUATIONS and INEQUALITIES in ONE VARIABLE
51

Linear Equations and Inequalities in One Variable

Nov 15, 2014

Download

Technology

 
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
  • 1. LINEAR EQUATIONS and INEQUALITIES in ONE VARIABLE

2. Linear equations and Inequalities in One Variable Equation and Inequalities are relationsbetween two quantities. 3.

  • Equationis a mathematical sentence indicating that two expressions are equal. The symbol = is used to indicate equality.
  • Ex.
  • 2x + 5 = 9is a conditional equation
  • since its truth or falsity depends onthe value of x
  • 2 + 9 = 11 is identityequation since both of itssides are identical to the samenumber 11.

4.

  • Inequalityis a mathematical sentence indicating that two expressions are not equal. The symbols , are used to denote inequality.
  • Ex.
  • 3 + 2 4 is an inequality
  • If two expressions are unequal, then their relationship can be any of the following, >, , < or .

5.

  • Linear equation in one variableis an equation which can be written in the form of ax + b = 0, where a and b are real-number constants and a 0.
  • Ex.
  • x + 7 = 12

6. Solution Set of a Linear Equation

  • Example
  • 4x + 2 = 10this statement is either true offalse
  • If x = 1, then4x + 2 = 10is false because 4(1) + 2 is 10
  • If x = 2, then4x + 2 = 10is true because 4(2) + 2 = 10

7. B. x 4 < 3this statement is either true or false If x =6, then x 4 is true because 6 4 < 3 If x = 10 , then x 4 is false because 6 4 is not < 3

  • When a number replaces a variable in an equation (or inequality) to result in a true statement, that number is asolutionof the equation (or inequality). The set of all solutions for a given equation (or inequality) as called the solution set of the equation (or inequality).

8. Solution Set of Simple Equations and Inequalities in One Variable by Inspection

  • To solve an equation of inequality means to find its solution set. There are three(3) ways to solve an equation or inequality by inspection

9. A. Guess-and-Check

  • In this method, one guesses and substitutes values into an equation of inequality to see if a true statement will result.

10. Consider the inequality x 12 < 4 If x = 18, then 18 12 is not < 4 If x = 17, then 17 12 is not < 4 If x = 16, then 16 12 is not < 4 If x = 15, then 15 12< 4 If x = 14, then 14 12< 4

  • Inequality x - 12 < 4 is true for all values of x which are less than 16. Therefore, solution set of the given inequality is x < 16.

11. Another example

  • X + 3 = 7
  • If x = 6, then 6 + 3 7
  • If x = 5, then 5 + 3 7
  • If x = 4, then 4 + 3 = 7
  • Therefore x = 4

12. B. Cover-up

  • In this method , one covers up the term with the variable.

13. Example

  • Consider equation x + 9 = 15
  • x + 9 = 15
  • + 9 = 15
  • To result in a true statement, themust be 6. Therefore x = 6

14.

  • Another example
  • X 1 = 3
  • 1 = 3
  • x = 4

15. C. Working Backwards

  • In this method, the reverse procedure is used

16. Consider the equation 2x + 6 = 4

  • timesequalsplusequals
  • 22x6
  • Start
  • 14End
  • 286
  • equalsdividedequalsminus

x 17. Example: 4y = 12

  • timesequals
  • 4
  • Start12 End
  • 4
  • equalsdividedTherefore y = 3

y 18. Properties of Equality and Inequality 19. Properties of Equality

  • Let a, b, and c be real numbers.
  • Reflexive Property
  • a = a
  • Ex. 3 = 3, 7 = 7 or 10.5 = 10.5

20. B. Symmetric Property

  • If a = b, then b = a
  • Ex. If 3 + 5 = 8, then 8 = 3 + 5
  • If 15 = 6 + 9, then 6 + 9 = 15
  • If 20 = (4)(5), then (4)(5) = 20

21. C. Transitive Property

  • If a = b and b = c, then a = c
  • Ex. If 8 + 5 = 13 and 13 = 6 + 7
  • then 8 + 5 = 6 + 7
  • If (8)(5) = 40 and 40 = (4)(10)
  • then (8)(5) = (4)(10)

22. D. Addition Property

  • Ifa = b, then a + c = b + c
  • Ex. If 3 + 5 = 8, then (3 + 5) = 3 = 8 +3

23. E. Subtraction Property

  • If a = b, then a c = b c
  • Ex. 3 + 5 = 8, then (3 + 5) 3 = 8 - 3

24. F. Multiplication Property

  • If a = b, then ac = bc
  • Ex. (4)(6) = 24, then (4)(6)(3) = (24)(3)

25. G. Division Property

  • If a = b, and c 0, then a/c = b/c
  • Ex. If (4)(6) = 24, then (4)(6)/3 =24/3

26. Properties of Inequality

  • Let a, b and c be real numbers.
  • Note: The properties of inequalities will still hold true using the relation symbol and .

27. A. Addition Property

  • If a < b, then a + c < b + c
  • Ex. If 2 < 3, then 2 + 1 < 3 + 1

28. B. Subtraction Property

  • If a < b, then a c < b c
  • Ex. If 2 < 3, then 2 1 < 3 1

29. C. Multiplication Property

  • If a < b and c > 0, then ac < bc
  • IF a < b and c < 0, then ac > bc
  • Ex. If 2 < 3, then (2)(2) < (3)(2)
  • If 2 < 3, then (2)(-2) > (3)(-2)

30. D. Division Property

  • If a < b and c > 0, then a/c < b/c
  • If a < b and c < 0, then a/c > b/c
  • Ex. If 2 < 3, then 2/3 < 3/3
  • If 2 < 3, then 2/-3 > 3/-3

31. Solving Linear Equations in One Variable 32.

  • Example:
  • Solve the following equations:
  • x 5 = 8
  • x 5 + 5 = 8 + 5add 5 to both sides
  • x + 0 = 13 of the equation
  • x = 13
  • Recall that if the same number is added to both sides of the equation, the resulting sums are equal.

33.

  • x 12 = -18
  • x 12 + 12 = -18 + 12add 12 to both sides
  • x + 0 = -6
  • x = -6
  • This problem also uses the addition property of equalities.

34.

  • x + 4 = 6
  • x + 4 4 = 6 4subtract 4 to both sides of
  • x + 0 = 2the equation
  • x = 2
  • Recall that if the same number is subtracted to both sides of the equation, the differences are equal.

35.

  • x + 12 = 25
  • x + 12 12 = 25 12subtract 12 to both
  • x + 0 = 25 12sides
  • This problem also uses the subtraction property of equalities.

36.

  • x/2 = 3
  • x/2 . 2 = 3 . 2multiply both sides by 2
  • x = 6
  • Recall that if the same number is multiplied to both sides of the equation, the products are equal.

37.

  • 6.x/7 = -5
  • x/7 . 7 = -5 .7multiplyboth sides by 7
  • x = -35
  • This problem also uses multiplication property of equalities.

38.

  • 7.5 x = 35
  • 5x/5 = 35/5both sides of the equation is
  • X = 7divided by the numericalcoefficient of x to makethe coefficient of x equals to 1
  • Recall the if both sides of the equation is divided by a non-zero number, the quotients are equal.

39.

  • 8.12y = -72
  • 12y/12 = -72/12divide both sides by 12
  • y = -6
  • This problem also uses the division property of equalities.

40.

  • Other equations in one variable are solved using more than on property of equalities.
  • 9.2x + 3 = 9
  • 2x+ 3 3 = 9 3subtraction property
  • 2x = 6
  • 2x/2 = 6/2division property
  • x = 3

41.

  • 10.5y 4 = 12 y
  • 5y 4 + 4 = 12 y + 4addition property
  • 5y = 16 y
  • 5y + y = 16 y + yaddition property
  • 6y = 16
  • 6y/5 = 16/5division property
  • y = 2 4/6

42. Solving Linear Inequalities in One Variable 43.

  • The solution set o inequalities maybe represented on a number line.
  • Recall that a solution of a linear inequality in one variable is a real number which makes the inequality true.
  • Example:
  • 1. Graph x > 6 on a number line
  • Ox>6
  • 01234567891011
  • The ray indicates the solution set of x > 6

44.

  • The ray indicates the that he solution set, x > 6 consist of all numbers greater than 6. The open circle of 6 indicates that 6 is not included.

45.

  • 2. Graph the solution set x -1 on a number line.
  • x -1
  • -2-101
  • The ray indicates that the solution set of x -1 consist of all the numbers less than or equal to-1. The solid circle of -1 indicates that -1 is included in the solution set.

46.

  • Applying the Properties of Inequalities in Solving Linear Inequalities:
  • 1. Solve x 2 > 6 and graph the solution set.
  • x 2 > 6
  • x 2 + 2 > 6 + 2add 2 to both sides of the
  • x + 0 > 8inequality
  • x > 8
  • Ox > 8
  • 8

47.

  • 2.x + 15 < -7
  • x + 15 15 < -7 15subtract 15 from bothsides of the
  • x +0 < - 22inequalities.
  • x < -22
  • x < -22o
  • -22

48. Solving Word Problems Involving Linear Equations 49.

  • Steps in solving word problems:
  • Read and understand the problem. Identify what is given and what is unknown. Choose a variable to represent the unknown number.
  • Express the other unknown, if there are any., in terms of the variable chosen in step 1.
  • Write a equation to represent the relationship among the given and unknown/s.
  • Solve the equation for the unknown and use the solution to find for the quantities being asked.
  • Check by going back to the original statement.

50.

  • Example:
  • One number is 3 less than another number. If their sum is 49, find the two numbers.
  • Step 1: Let x be the first number.
  • Step 2: Let x 3 be the second number.
  • Step 3: x + ( x 3) = 49
  • Step 4: x + x 3 = 49
  • 2x 3 = 49
  • 2x = 49 + 3
  • 2x = 52
  • x = 26 the first number
  • x 3 = 23the second number
  • Step 5: Check: The sum of 26 and 23 is 49,
  • and 23 is 3 less than 26.

51.

  • 2. Six years ago, Mrs. dela Rosa was 5 times as old as her daughter Leila.
  • How old is Leila now if her age is one-third of her mothers present age?
  • Solution:
  • Step 1: Let x be Leilas age now
  • 3x is Mrs. dela Rosas age now
  • Step 2: x 6 is Leilas age 6 years ago
  • 3x 6 is Mrs. dela Rosas age 6 years ago
  • Step 3: 5(x 6) = 3x 6
  • Step 4: 5(x 6) = 3x 6
  • 5x 30 = 3x 6
  • 5x 30 + 30 = 3x 6 + 30
  • 5x = 3x + 24
  • 5x 3x = 3x +24 3x
  • 2x = 24
  • 2x/2 = 24/ 2
  • X = 12Leilas age now
  • 3x = 36Mrs. dela Rosas age now
  • Step 5: Check: Thrice of Leilas present age, 12, is Mrs. dela Rosas presnt age, 36. Six years ago, Mrs. dela Rosa was 36 6 = 30years old which was five times Leilas age, 12 6 = 6.