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Math 374

Math 374 . Inequalities. Topics Covered. 1) Number Lines 2) Inequality Sign 3) Inequality Form 4) Interval Form. Number Lines. People have always turned to pictures to help them visualize concepts. The concept of numbers has traditionally been represented by a line.

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Math 374

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  1. Math 374 Inequalities

  2. Topics Covered • 1) Number Lines • 2) Inequality Sign • 3) Inequality Form • 4) Interval Form

  3. Number Lines • People have always turned to pictures to help them visualize concepts. • The concept of numbers has traditionally been represented by a line. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  4. Number Lines Notes • We consider zero to be the centre with no sign • We say positive extends to the right and negative extends to the left -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  5. Number Line Notes • The negatives are a mirror image of the positives and vice versa • We say the numbers continue to the right to positive infinity represented by + ∞ • Numbers continue to the left to negative infinity represented by - ∞ • We can represent all the numbers that we know this way

  6. Notes • Consider ¾ ¾ 0 1 Enclosing Integers How many numbers are between 0 & 1? The answer is ∞

  7. Notes • What is the next number after 1? • The answer is impossible to determine. • Consider… 5 6 7 • We say 5 is less than 7 because 5 falls to the left of 7. We need a symbol.

  8. Notes • < or > • The symbol has two sides • Remember that the direction that the sign “hugs” is bigger. • Thus 5 < 7 (5 is less than 7) • 7 > 5 (7 is larger than 5) • The two statements mean the same but use different symbols

  9. Notes • To make this work, we will always work by reading left to right • We want to handle this like equations • Let us look at what we do • 5 < 7 • (Adding 5) 5 + 5 < 7 + 5 • 10 < 12 (still true)

  10. Notes • (Subtract 5) 5 – 5 < 7 – 5 • 0 < 2 … still true • This operation is called transposing and so this works! • (Multiply 5) 5 x 5 < 7 x 5 • 25 < 35 still true … what is next?

  11. Notes • (multiply by -5) 5 x (-5) < 7 x (-5) • -25 < -35 is FALSE • To make it true, we must reverse the inequality sign. • Therefore -25 > -35 now true

  12. Notes • (Divide by 2) 5 / 2 < 7 / 2 • 2.5 < 3.5 still true • (divide by -2) 5 / -2 < 7 / -2 • -2.5 < - 3.5 FALSE • To make it true we must reverse the inequality sign. Thus – 2.5 > - 3.5

  13. Final Important Notes • 1) We will always move our x to the left side • 2) Whenever we multiply or divide by a negative number, we reverse the inequality sign • Additional symbols • ≤ Greater than or equal to • ≥ Less than or equal to Filled means included Unfilled means not included

  14. Exercises • 5x – 4 ≥ 2x + 11 • 3x ≥ 15 • x ≥ 5 (This is inequality form) • 4x – 7 < 8x + 9 • -4x < 16 • x > -4

  15. Other Forms • Graph Form • We reproduce a miniature number line • Ex x ≥ 5 - ∞ + ∞ 5

  16. Notes • Ex x < - 3 • Ex. x > ¾ Note ¾ = 0.75 • Ex x ≤ -42 3 • Note that this is -21.5 -3

  17. Notes • Intervals and Gaps • We can work on the number line and create some strange situations. The key are the end points Interval Form 3 8

  18. Notes • This is a GRAPH FORM INTERVAL • We have all the numbers between 3 and 8 including 8 but not including 3. • Using inequality form… • 3 < x ≤ 8 • Using Interval Form 3 8

  19. Notes • Interval Form • ]3, 8] • Think hugging and not hugging. If it does not hug the 3, it does not include the 3. • If it hugs the 8, it includes the 8. • You can never equal ∞

  20. Notes • Consider • - ∞ ∞ Graph Form • Inequality Form… x ≥ 6 • Interval Form [6, ∞ [ • Do #1 a – y (a, v & y as examples) 6

  21. Notes • Given one form, give other forms… • Consider ] - ∞, - 7 ] Interval form • Graph form • Inequality form x < - 7 -7

  22. New Symbol! • Consider x ≤ - 5 U x > 7 (U means union) • State interval form -5 7

  23. Notes • State the Interval form • - ∞, 5 ] U ] 7, ∞ • Consider 6x – 5 ≥ 9x + 5 • -3x ≥ 10 • X ≤ -10 3 ] -∞, - 10 3

  24. Notes • In graph form… • Do #2 a – j ; #3 a j -10 3 -3 -4

  25. Standard Form • Consider 5y – 3x ≥ 2 • 5y ≥ 3x + 2 • y ≥ 3x + 2 5 5

  26. Standard Form • 9x – 7y – 100 < 0 • - 7y < -9x + 100 • y > - 9x + 100 -7 - 7 y > 9x – 100 7 7 Notice the inequality changed signs AND notice what happened to the sign because a + divided by a – becomes -. Do #4 a - j

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