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- 1. INEQUALITIES
- 2. INTRODUCTION If a and b are real numbers then we can compare their positions by the relation… Less than < Greater than > Less than or equal to ≤ Greater than or equal to ≥ For example: if x > 3 , it means x can be any value more than 3
- 3. <ul><li>VARIOUS TYPES </li></ul><ul><li>OF </li></ul><ul><li>GRAPHS </li></ul>Shade up Shade down Solid line Dashed line > < WHERE :
- 4. y = x 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y = x
- 5. y x 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y x
- 6. y x 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y x
- 7. y = x + 1 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y = x + 1
- 8. y x + 1 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y x + 1
- 9. y x + 1 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y x + 1
- 10. y > x + 1 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y > x + 1
- 11. y < x + 1 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 y < x + 1
- 12. x > 2 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 x > 2
- 13. WRITE THE EQUATION 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 4
- 14. WRITE THE EQUATION 1 2 3 – 1 – 2 – 3 3 2 1 – 1 – 2 – 3 4
- 15. PROPERTIES OF INEQUALITIES <ul><li>If ‘a’ is greater than ‘b’ </li></ul>If we add ‘c’ (any real number) then which one is greater A + c or b + c Solution : (a + c) is greater than b + c
- 16. You know 8 is greater than 4 or 8 > 4 Add 2 on both sides 8 + 2 > 4 + 2 10 > 6 TRUE EXAMPLE
- 17. PROPERTIES OF INEQUALITIES If ‘a’ is greater than ‘b’ If we subtract ‘c’ (any real number) then which one is greater a – c or b – c Solution: (a – c) is greater than b – c
- 18. You know 8 is greater than 4 or 8 > 4 Subtract 2 from both sides 8 – 2 > 4 – 2 6 > 2 TRUE EXAMPLE
- 19. If ‘a’ is greater than ‘b’ i.e. (a > b) If we multiply by ‘c’ (any real number) then which one is greater ac or bc ? Depends upon ‘c’ because c can be a positive or negative real number PROPERTIES OF INEQUALITIES
- 20. You know 8 is greater than 4 or 8 > 4 Multiply by 2 both sides 8(2) > 4(2) 16 > 8 TRUE EXAMPLE
- 21. You know 8 is greater than 4 or 8 > 4 Multiply by – 2 both sides 8(– 2) > 4(– 2) – 16 > – 8 FALSE EXAMPLE
- 22. REMEMBER If ‘a’ is greater than ‘b’ WHICH IS GREATER ac or bc If c is positive then ac > bc If c is negative then ac < bc
- 23. If ‘a’ is greater than ‘b’ WHICH IS GREATER or Is Greater than REMEMBER
- 24. REMEMBER We know that 8 is greater than 4 or 8 > 4 Take reciprocals on both sides
- 25. <ul><li>Transitive : If a < b and b < c, then a < c </li></ul><ul><li>Addition of inequalities: If a < b and </li></ul><ul><li>c < d, then a + c < b + d. </li></ul><ul><li>Addition of a constant: If a < b, then </li></ul><ul><li>a + c < b + c. </li></ul><ul><li>Multiplication by a constant: </li></ul><ul><li>If a < b and c is positive real number, then: ac < bc </li></ul><ul><li>and if c is negative real number, then ac > bc </li></ul><ul><li>Taking Reciprocals: If a < b and a, b ≠ 0 , then </li></ul>PROPERTIES OF INEQUALITIES
- 26. INTERVAL NOTATION Real number line graph Inequality notation
- 27. SOLVING LINEAR INEQUATIONS You solve linear inequalities in the same way as you would solve linear equations, but with one exception. Property: If in the process of solving an inequality, you multiply or divide the inequality by a negative number, then , you must switch the direction of the inequality. If – x > a , then x < – a .
- 28. Solve x + 3 < 2. <ul><li>Graphically </li></ul><ul><li> </li></ul>SOLVING LINEAR INEQUATIONS CASE-1 When the equation was " x + 3 = 2“ type, We normally subtract 3 from both sides. Then the solution is: x < –1 X + 3 < 2 - 3 -3 -------------------------
- 29. Solve 2 x < 9. <ul><li>Like inequality divide by 2 </li></ul>CASE-2
- 30. What happens when the number is negative? If you divide both sides by – 3, (The inequality will change if we multiply or divide with a negative number on both sides.) CASE-3
- 31. Solve the followings: 1. 3x – 4 > 8 2. – 6x – 18 < – 24 Answers: 1. x > 4 2. x > 1 EXAMPLE
- 32. SOLVING QUADRATIC INEQUATIONS When we have an inequality with " x 2 " as the highest-degree term, it is called a "quadratic inequality". Solve x 2 – 3 x + 2 > 0 Step 1: Change the inequality to an equation . Find x- interce p t x 2 – 3 x + 2 = 0 (x – 1) (x – 2) = 0 x = 1 or 2 Step 2 : Plot the points ( x = 1, 2) on the number line The number line is divided into the intervals (- ∞ , 1), (1, 2), and (2, ∞). 3 2 1 0 1 2 3 1 2
- 33. x 2 – 3 x + 2 > 0 or (x – 1) (x – 2) > 0 The number line is divided into the intervals (- ∞ , 1), (1, 2), and (2, ∞). Test-point method: Pick a point (any point) in each interval x = 3 (x - 1) is p ositive (x - 1) (x - 2) is positive (x - 3) is p ositive POSITIVE x = 1.5 (x - 1) is negative (x - 1) (x - 2) is negative (x - 3) is p ositive NEGATIVE x = 0 (x - 1) is negative (x - 1) (x - 2) is positive (x - 2) is negative POSITIVE (x – 1) (x – 2) is positive when x > 2 or x < 1 SOLVING QUADRATIC INEQUATIONS 1 2 1 2
- 34. WHICH ONE IS GREATER? x 2 x or You can’t say Lets find the interval where x 2 is greater than x
- 35. SOLUTION For what value of x ? x 2 – x ≥ 0 or (x) (x – 1) ≥ 0 Step 1: Change the inequality to an equation . Find value of x x = 0, 1 x 2 – x = 0 Step 2: Plot the points Step 3: Test point method At x = 2 At x = 0.5 At x = -1 + ─ + Step 4: Solution x > 1 or x < 0 0 1 0 1 0 1
- 36. You can solve some absolute-value equations using logics. For instance, you have learned that the equation | x | 8 has two solutions: 8 and 8. SOLVING ABSOLUTE-VALUE EQUATIONS To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative. Because I X I = + X if X > 0 - X If X < 0 0 if X = 0
- 37. SOLVING AN ABSOLUTE-VALUE EQUATION Solve | x 2 | 5 x 2 IS POSITIVE | x 2 | 5 x 7 x 3 x 2 IS NEGATIVE | x 2 | 5 | 7 2 | | 5 | 5 | 3 2 | | 5 | 5 The expression x 2 can be equal to 5 or 5 . x 2 5 Solve | x 2 | 5 S OLUTION x 2 5 The equation has two solutions: 7 and –3. x 2 IS POSITIVE x 2 5 The expression x 2 can be equal to 5 or 5 . x 2 IS POSITIVE | x 2 | 5 x 2 5 x 7 x 2 IS POSITIVE | x 2 | 5 x 2 5 x 7 x 2 IS NEGATIVE x 2 5 x 3 x 2 IS NEGATIVE | x 2 | 5 x 2 5 C HECK
- 38. Recall that x is the distance between x and 0. If x 8, then any number between 8 and 8 is a solution of the inequality. You can use the following properties to solve absolute-value inequalities and equations. Recall that | x | is the distance between x and 0. If | x | 8, then any number between 8 and 8 is a solution of the inequality. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 Recall that | x | is the distance between x and 0. If | x | 8, then any number between 8 and 8 is a solution of the inequality.
- 39. a x b c and a x b c. a x b c and a x b c. a x b c or a x b c. a x b c or a x b c. a x b c or a x b c. | a x b | c | a x b | c | a x b | c | a x b | c | a x b | c When an absolute value is less than a number, the inequalities are connected by and . When an absolute value is greater than a number, the inequalities are connected by or . S OLVING A BSOLUTE- V ALUE I NEQUALITIES S OLVING A BSOLUTE- V ALUE E QUATIONS AND I NEQUALITIES means means means means means means means means means means
- 40. Solve | x 4 | < 3 x 4 IS POSITIVE x 4 IS NEGATIVE | x 4 | 3 x 4 3 x 7 | x 4 | 3 x 4 3 x 1 Reverse inequality symbol. This can be written as 1 x 7 . The solution is all real numbers greater than 1 and less than 7. EXAMPLE
- 41. 2 x 1 9 | 2 x 1 | 3 6 | 2 x 1 | 9 2 x 10 2 x + 1 IS NEGATIVE x 5 Solve | 2 x 1 | 3 6 and graph the solution. | 2 x 1 | 3 6 | 2 x 1 | 9 2 x 1 +9 2 x 8 2 x + 1 IS POSITIVE x 4 SOLVING AN ABSOLUTE-VALUE INEQUALITY Reverse inequality symbol. The solution is all real numbers greater than or equal to 4 or less than or equal to 5 . This can be written as the compound inequality x 5 or x 4 . 5 4 . | 2 x 1 | 3 6 | 2 x 1 | 9 2 x 1 +9 x 4 2 x 8 | 2 x 1 | 3 6 | 2 x 1 | 9 2 x 1 9 2 x 10 x 5 2 x + 1 IS POSITIVE 2 x + 1 IS NEGATIVE 6 5 4 3 2 1 0 1 2 3 4 5 6
- 42. SOLVING THE INEQUALITIES WITH THE HELP OF OPTIONS We can solve all the inequality questions by going with the options. Take an example: x 2 – 7x + 10 < 0 (1) X < 2 (2) x > 5 (3) x < 5 (4) 2 < x < 5 (5) Both (1) and (2)
- 43. SOLUTION Since the first option is x < 2, we take x = 1 and check whether the given inequality is satisfying or not. If x = 1, 1 2 – 7(1) + 10 < 0 4 < 0 (wrong) Option (1), (3) and (5) are wrong. Now take x = 6, 6 2 – 7 6 + 10 < 0 4 < 0 (wrong) So, option (2) is wrong. So, the answer is (4).
- 44. QUANTITATIVE COMPARISON QUESTIONS <ul><li>2 < x < 3 and - 6 < y < - 5 </li></ul><ul><li>Which is greater? </li></ul><ul><li>x 4 y or xy 4 </li></ul>Column A will always be negative where B is always positive.
- 45. EXAMPLE <ul><li>If |x – 3| > 2, which will be greater? </li></ul><ul><li>Column A Column B </li></ul><ul><li>|x| 2 </li></ul>|x – 3| > 2 means x > 5 or x < 1 If x > 5, |x| > 2 But if x < 1, we can’t say

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