Math 7 inequalities and intervals

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Math 7 inequalities and intervals

  1. 1. Let’s Start!
  2. 2. Inequalities & Intervals A Mathematics 7 Lecture
  3. 3. What’s an inequality? • It is a range of values, rather than ONE set number • It is an algebraic relation showing that a quantity is greater than or less than another quantity. Inequalities and Intervals
  4. 4.     Less than Greater than Less than OR EQUAL TO Greater than OR EQUAL TO Inequality Symbols Inequalities and Intervals
  5. 5. True or false? Inequality Symbols 5 4 3 2 5 4   6.5 6.4   3 3 3 2 5 6   3 3 True False False True False True False True Inequalities and Intervals
  6. 6. INEQUALITIES AND KEYWORDS < >   •less than •fewer than •greater than •more than •exceeds •less than or equal to •no more than •at most •greater than or equal to •no less than •at least Keywords Inequalities and Intervals
  7. 7. Keywords Examples Write as inequalities. 1. A number x is more than 5 x > 5 2. A number x increased by 3 is fewer than 4 x + 3 < 4 3. A number x is at least 10 x  10 4. Three less than twice a number x is at most 7 2x  3  7 Inequalities and Intervals
  8. 8. Recall: order on the number line On a number line, the number on the right is greater than the number on the left. If a and b are numbers on the number line so that the point representing a lies to the left of the point representing b, then a < b or b > a. Graphs of Inequalities Inequalities and Intervals
  9. 9. Given a real number a and any real number x: Graphs of Inequalities all values of x to the LEFT of a x > ax < a a all values of x to the RIGHT of a The point is has a hole because a is excluded Inequalities and Intervals
  10. 10. Given a real number a and any real number x: Graphs of Inequalities all values of x to the LEFT of a, INCLUDING a x  ax  a a all values of x to the RIGHT of a, INCLUDING a The point is shaded because a is included Inequalities and Intervals
  11. 11. Given a real number a and any real number x: Graphs of Inequalities a darken the part of the number line that is to the RIGHT of the constant Place a point with a HOLE at x = a x a Inequalities and Intervals
  12. 12. Place a point with a HOLE at x = a Given a real number a and any real number x: Graphs of Inequalities a darken the part of the number line that is to the LEFT of the constant x a Inequalities and Intervals
  13. 13. Given a real number a and any real number x: Graphs of Inequalities a darken the part of the number line that is to the RIGHT of the constant Place a point with a SHADE at x = a x a Inequalities and Intervals
  14. 14. Place a point with a SHADE at x = a Given a real number a and any real number x: Graphs of Inequalities a darken the part of the number line that is to the LEFT of the constant x a Inequalities and Intervals
  15. 15. Given a real number a and any real number x: Graphs of Inequalities a darken the part of the number line that is to the RIGHT of the constant Place a point with a SHADE at x = a x a Inequalities and Intervals
  16. 16. Graphs of Inequalities Let’s compare graphs! Inequalities and Intervals
  17. 17. Examples Graphs of Inequalities x < 0 x > 2 Inequalities and Intervals
  18. 18. Examples Graphs of Inequalities Linear Inequalities x  5 x  3
  19. 19. Check your understanding Sketch the graph of the following inequalities on a number line. 1. x > 6 2. x  7 3. x  1 4. x < 8
  20. 20. Check your understanding State the inequality represented by the given graph. 1. 2. 3. 4.
  21. 21. • These are also called double inequalities. • These inequalities represent “betweeness” of values; i.e., values between two real numbers Compound Inequalities Inequalities and Intervals
  22. 22. Compound Inequalities Linear Inequalities a x b  x is between a and b x is greater than a and less than b a x b  x is between a and b inclusive x is greater than or equal to a and less than or equal to b
  23. 23. Compound Inequalities We can also have: a x b  x is greater than a and less than or equal to b a x b  x is greater than or equal to a and less than b Inequalities and Intervals
  24. 24. Compound Inequalities Graphs a x b  a x b  a x b  a x b  Inequalities and Intervals
  25. 25. Example: Compound Inequalities This inequality means that x is BETWEEN 2 and 3 2 3x   This also means that x is GREATER than 2 and LESS THAN 3 Inequalities and Intervals
  26. 26. Example: Compound Inequalities Linear Inequalities 2 3x  
  27. 27. Example: Compound Inequalities 2 3x   This inequality means that x is BETWEEN 2 and 3 INCLUSIVE Inequalities and Intervals
  28. 28. Example: Compound Inequalities 2 3x   2 3x   Inequalities and Intervals
  29. 29. Check your understanding Sketch the graph of the following inequalities on a number line. 1. 5 < x < 5 2. 4  x  7 3.  3 < x  1 4. 2  x < 8
  30. 30. Interval Notation • The set of all numbers between two endpoints is called an interval. • An interval may be described either by an inequality, by interval notation, or by a straight line graph. • An interval may be: – Bounded: • Open - does not include the endpoints • Closed - does include the endpoints • Half-Open - includes one endpoint – Unbounded: one or both endpoints are infinity Inequalities and Intervals
  31. 31. Notations • A parenthesis ( ) shows an open (not included) endpoint • A bracket [ ] shows a closed [included] endpoint • The infinity symbol () is used to describe very large or very small numbers + or  - all numbers GREATER than another  - all numbers GREATER than another Note that “” is NOT A NUMBER! Interval Notation Inequalities and Intervals
  32. 32. Interval Notation INEQUALITY SET NOTATION INTERVAL NOTATION x > a { x | x > a } (a, +) x < a { x | x < a } (-, a) x  a { x | x  a } [a, +) x  a { x | x  a } (-, a] Inequalities and Intervals Unbounded Intervals
  33. 33. Interval Notation INEQUALITY SET NOTATION INTERVAL NOTATION a < x < b { x | a < x < b } (a, b) a  x  b { x | a  x  b } [a, b] a < x  b { x | a < x  b } (a, b] a  x < b { x | a  x < b } [a, b) Bounded Intervals Inequalities and Intervals
  34. 34. Interval Notation Example: This represents all numbers GREATER THAN OR EQUAL TO 1  1,  In inequality form, this is x  1 Inequalities and Intervals
  35. 35. Interval Notation Example: The symbol before the –1 is a square bracket which means “is greater than or equal to." The symbol after the infinity sign is a parenthesis because the interval goes on forever (unbounded) and since infinity is not a number, it doesn't equal the endpoint (there is no endpoint). Inequalities and Intervals  1, 
  36. 36. Interval Notation Example: Write the following inequalities using interval notation 2x   2, 2x   ,2 2x   2, 2x   ,2 Inequalities and Intervals
  37. 37. Interval Notation Example: Write the following inequalities using interval notation 0 2x   0,2 0 2x   0,2 0 2x   0,2 0 2x   0,2 Inequalities and Intervals
  38. 38. Interval Notation Example: Write the following inequalities using interval notation 0 2x   0,2 0 2x   0,2 0 2x   0,2 0 2x   0,2 Inequalities and Intervals
  39. 39. Interval Notation Example: Graph the following intervals: (, 0) [3, +) Inequalities and Intervals
  40. 40. Interval Notation Linear Inequalities Example: Graph the following intervals:  2,3  1,6
  41. 41. Check your understanding Write the following using interval notation, then sketch the graph. 1. 1 < x < 1 2. 4  x < 7 3. x  2 4. x > 6
  42. 42. Thank you!

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