Linear Equations and Inequalities in One Variable

51
LINEAR EQUATIONS and INEQUALITIES in ONE VARIABLE

description

 

Transcript of Linear Equations and Inequalities in One Variable

  • 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.