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Solving Inequalities with Graphs

Learn to solve inequalities, graph results accurately, understand inequality words, and tackle word problems confidently. The only difference from regular equations is how negatives are handled. Master inequality solving through step-by-step examples and problem-solving techniques.

johndlewis
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Solving Inequalities with Graphs

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  1. Unit 4 Inequalities • Graphing Inequalities • Translating Inequality Words • Solving Inequalities • Inequality Word Problems

  2. Inequalities OBJECTIVE: Given five inequalities, the student will be able to solve for the variable and graph the result with 80% accuracy. Ca.SS 5.0: Students solve multi-step equations involving linear inequalities.

  3. 1. 5y + 8 < -92 2. -7 – 10y 5 – 7y 3. 4. 0.6x – 1.8  1.2 x Check for Understanding

  4. = Inequalities

  5. = NOT Inequalities

  6. Larger # Smaller # Smaller # Larger # Inequalities Greater Than Less Than

  7. ≥ Greater than or equal > Greater than ≤ Less than or equal <Less than  ≥ -7 -6 -5 -4 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10  >  ≤  < Graphing Inequalities X ≥ 1 X > 5 X ≤ -3 X < 0

  8. X ≥ 8 -7 -3 -3 The Only Way They Are Different From Regular Equations is… If there is a negative coefficient, then you ; • Treat the inequality exactly the same as you would solve an equation • Then change the direction of the “inequality” on the last step. -3X < -18 (-7) (-7) X > 6 X ≤ -56

  9. 8y –4 < -52 +4 +4 8y < -48 8 8 y < -6 15x + 3–14x 7 x + 3  7 -3 -3 x  4 Try These… 8y –4 < -52 15x + 3 –14x 7

  10. 6 – 8m> 22 -6 -6 - 8m > 16 -8 -8 m < -2 7 – 8y 5 – 7y +7y +7y 7 – x  5 -7 -7 - x  -2 -1 -1 x 2 Try These… 6 – 8m> 22 7 – 8y 5 – 7y

  11. 10 – 3x  4 -10 -10 -3x  -6 -3 -3 x  2 Try These… 5 3 1 •15 • 15 • 15

  12. 0.3x + 5.4  -1.0x + 0.2 Move every decimal over 1 place 3x + 54  -10x + 2 +10x +10x 13x + 54  2 -54 -54 13x  -52 13 13 x  -4 Try These… 0.3x+5.4  -x+0.2

  13. 1. 5y + 8 < -92 2. -7 – 10y 5 – 7y 3. 4. 1.2 x  0.6x – 1.8 Check for Understanding y< -20 y -3 x 1 x -3

  14. Summary Inspiration: • What do the following inequalities mean? >  ≤  • What do you do when you multiply or divide by a negative? • How is solving inequalities like solving equations?

  15. Writing Inequalities OBJECTIVE: Given 5 word problems, the student will be able to write the inequality with 80% accuracy. Ca.SS 5.0: Students solve multi-step equations involving linear inequalities.

  16. Check for Understanding • A number is less than 7 • 5 more than a number is at least 3 • 13 less than x is more than 5 • Twice a number m is at most –4 • The number x is at least as great as the number y

  17. You always need to think about $$$ in your pocket The approach… If I have at least 5 dollars in my pocket…

  18. You always need to think about $$$ in your pocket The approach… Then I have 5 dollars or more (greater than or =)

  19. You always need to think about $$$ in your pocket The approach… If I have at most 30 dollars in my pocket Then I have 5 dollars or more (greater than or =)

  20. You always need to think about $$$ in your pocket The approach… Then I have 30 dollars or less (less than or =) Then I have 5 dollars or more (greater than or =)

  21. You always need to think about $$$ in your pocket The approach… Then I have 30 dollars or less (less than or =) Then I have 5 dollars or more (greater than or =) If I have more than 50 dollars…

  22. You always need to think about $$$ in your pocket The approach… Then I have 30 dollars or less (less than or =) Then I have 5 dollars or more (greater than or =) Then I have $50.01 or more (greater than) If I have less than 17 dollars… By Marianne Ilmanen

  23. You always need to think about $$$ in your pocket The approach… Then I have 30 dollars or less (less than or =) Then I have 5 dollars or more (greater than or =) Then I have $50.01 or more (greater than) Then I have $16.99 or less (less than) By Marianne Ilmanen

  24. Inequalities in Word Problems A number decreased by negative 4 is at least 9. A numberdecreased by negative 4is at least 9. X – (-4) ≥ 9 X +4≥ 9

  25. At the maximum $9 At most $9 Greater than $9 At the minimum $9 Less than $9 At least $9 No more than $9 No less than $9 X  9 The Amount of Money in your Pocket is…. • X  9 • X  9 X  9 Interpretation Trick • X  9 • X  9 • X  9 X  9

  26. Let’s Try a Few A number “Y” is less than 4

  27. Let’s Try a Few Y < 4 A number “Y” is less than 4

  28. Let’s Try a Few Y < 4 A number “Y” is less than 4 A number “X” is greater than or equal to 2 ½

  29. Let’s Try a Few Y < 4 A number “Y” is less than 4 A number “X” is greater than or equal to 2 ½ X  2 ½

  30. Let’s Try a Few Y < 4 A number “Y” is less than 4 A number “X” is greater than or equal to 2 ½ X  2 ½ A number “M” is at least 3

  31. Let’s Try a Few Y < 4 A number “Y” is less than 4 A number “X” is greater than or equal to 2 ½ X  2 ½ A number “M” is at least 3 M  3 "At least" means equal or more...

  32. Let’s Try a Few A number “P” is at most –4

  33. Let’s Try a Few P  -4 A number “P” is at most –4

  34. Let’s Try a Few P  -4 A number “P” is at most –4 3 less than a number is at least 10

  35. Let’s Try a Few P  -4 A number “P” is at most –4 3 less thana numberis at least 10 X – 3 10

  36. Let’s Try a Few P  -4 A number “P” is at most –4 3 less thana numberis at least 10 X – 3 10 6 times a number is greater than - 1

  37. Let’s Try a Few P  -4 A number “P” is at most –4 3 less thana numberis at least 10 X – 3 10 6 timesa numberis greater than - 1 6m  -1

  38. Let’s Try a Few 13 less than twice a number is at least –6

  39. Let’s Try a Few 2x – 13 -6 13 less than twice a number is at least –6

  40. Let’s Try a Few 2x – 13 -6 13 less than twice a number is at least –6 7 more than half of a number is at most 5

  41. Let’s Try a Few 2x – 13 -6 13 less than twice a number is at least –6 ½ x + 7 5 7 more than half ofa numberis at most 5

  42. Let’s Try a Few 7 more than a number is more than –17

  43. Let’s Try a Few X + 7 -17 7 morethana numberis more than –17

  44. Let’s Try a Few X + 7 -17 7 morethana numberis more than –17 -15 less than “P” cubed is at least 35

  45. Let’s Try a Few X + 7 -17 7 morethana numberis more than –17 -15 less than“P” cubedis at least 35 P3+ 15  35

  46. Check for Understanding X< 7 X + 5  3 • A number is less than 7 • 5 more than a number is at least 3 • 13 less than x is more than 5 • Twice a number m is at most –4 • The number x is at least as great as the number y X – 13  5 2m -4 X Y

  47. Summary Inspiration • Describe the “trick” to interpreting the correct inequality from word problems. • What do inequalities mean? • How do Inequalities apply to your reality?

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