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Lesson 12: Properties of Inequalities Date: 11/14/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Lesson 12: Properties of Inequalities
Student Outcomes
Students justify the properties of inequalities that are denoted by < (less than), β€ (less than or equal), > (greater than), and β₯ (greater than or equal).
Classwork
Opening Exercise (10 minutes)
Students complete a two round sprint exercise where they practice their knowledge of solving linear equations in the form ππ₯ + π = π and π(π₯ + π) = π. Provide one minute for each round of the sprint. Follow the established protocol for a sprint exercise. Be sure to provide any answers not completed by the students.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Sprint β Round 1 Write the solution for each equation as quickly and accurately as possible within the allotted time.
1. π₯ + 1 = 5 23. 17π₯ = 5
2. π₯ + 2 = 5 24. 27π₯ = 10
3. π₯ + 3 = 5 25. 37π₯ = 15
4. π₯ + 4 = 5 26. 47π₯ = 20
5. π₯ + 5 = 5 27. β 57 π₯ = β25
6. π₯ + 6 = 5 28. 2π₯ + 4 = 12
7. π₯ + 7 = 5 29. 2π₯ + 5 = 13
8. π₯ β 5 = 2 30. 2π₯ + 6 = 14
9. π₯ β 5 = 4 31. 3π₯ + 6 = 18
10. π₯ β 5 = 6 32. 4π₯ + 6 = 22
11. π₯ β 5 = 8 33. βπ₯ β 3 = β10
12. π₯ β 5 = 10 34. βπ₯ β 3 = β8
13. 3π₯ = 15 35. βπ₯ β 3 = β6
14. 3π₯ = 12 36. βπ₯ β 3 = β4
15. 3π₯ = 6 37. βπ₯ β 3 = β2
16. 3π₯ = 0 38. βπ₯ β 3 = 0
17. 3π₯ = β3 39. 2(π₯ + 3) = 4
18. β9π₯ = 18 40. 3(π₯ + 3) = 6
19. β6π₯ = 18 41. 5(π₯ + 3) = 10
20. β3π₯ = 18 42. 5(π₯ β 3) = 10
21. β1π₯ = 18 43. β2(π₯ β 3) = 8
22. 3π₯ = β18 44. β3(π₯ + 4) = 3
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
ROUND 1 KEY
1. π + π = π π 23. πππ = π ππ
2. π + π = π π 24. πππ = ππ ππ
3. π + π = π π 25. πππ = ππ ππ
4. π + π = π π 26. πππ = ππ ππ
5. π + π = π π 27. βπππ = βππ ππ
6. π + π = π βπ 28. ππ + π = ππ π
7. π + π = π βπ 29. ππ + π = ππ π
8. π β π = π π 30. ππ + π = ππ π
9. π β π = π π 31. ππ + π = ππ π
10. π β π = π ππ 32. ππ + π = ππ π
11. π β π = π ππ 33. βπ β π = βππ π
12. π β π = ππ ππ 34. βπ β π = βπ π
13. ππ = ππ π 35. βπ β π = βπ π
14. ππ = ππ π 36. βπ β π = βπ π
15. ππ = π π 37. βπ β π = βπ βπ
16. ππ = π π 38. βπ β π = π βπ
17. ππ = βπ βπ 39. π(π + π) = π βπ
18. βππ = ππ βπ 40. π(π + π) = π βπ
19. βππ = ππ βπ 41. π(π + π) = ππ βπ
20. βππ = ππ βπ 42. π(π β π) = ππ π
21. βππ = ππ βππ 43. βπ(π β π) = π βπ
22. ππ = βππ βπ 44. βπ(π + π) = π βπ
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Sprint β Round 1 Write the solution for each equation as quickly and accurately as possible within the allotted time.
1. π₯ + 7 = 9 23. 15π₯ = 10
2. π₯ + 6 = 9 24. 25π₯ = 20
3. π₯ + 5 = 9 25. 35π₯ = 30
4. π₯ + 4 = 9 26. 45π₯ = 40
5. π₯ + 3 = 9 27. 55π₯ = 50
6. π₯ + 2 = 9 28. 3π₯ + 2 = 14
7. π₯ + 1 = 9 29. 3π₯ + 3 = 15
8. π₯ β 8 = 2 30. 3π₯ + 4 = 16
9. π₯ β 8 = 4 31. 2π₯ + 4 = 12
10. π₯ β 8 = 6 32. π₯ + 4 = 8
11. π₯ β 8 = 8 33. β2π₯ β 1 = 0
12. π₯ β 10 = 10 34. β2π₯ β 1 = 2
13. 4π₯ = 12 35. β2π₯ β 1 = 4
14. 4π₯ = 8 36. β2π₯ β 1 = 6
15. 4π₯ = 4 37. β2π₯ β 1 = 7
16. 4π₯ = 0 38. β2π₯ β 1 = 8
17. 4π₯ = β4 39. 3(π₯ + 2) = 9
18. β8π₯ = 24 40. 4(π₯ + 2) = 12
19. β6π₯ = 24 41. 5(π₯ + 2) = 15
20. β3π₯ = 24 42. 5(π₯ β 2) = β5
21. β2π₯ = 24 43. β3(2π₯ β 1) = β9
22. 6π₯ = β24 44. β5(4π₯ + 1) = 15
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Round 2 KEY
1. π + π = π π 23. πππ = ππ ππ
2. π + π = π π 24. πππ = ππ ππ
3. π + π = π π 25. πππ = ππ ππ
4. π + π = π π 26. πππ = ππ ππ
5. π + π = π π 27. πππ = ππ ππ
6. π + π = π π 28. ππ + π = ππ π
7. π + π = π π 29. ππ + π = ππ π
8. π β π = π ππ 30. ππ + π = ππ π
9. π β π = π ππ 31. ππ + π = ππ π
10. π β π = π ππ 32. π + π = π π
11. π β π = π ππ 33. βππ β π = π βππ
12. π β ππ = ππ ππ 34. βππ β π = π βππ
13. ππ = ππ π 35. βππ β π = π βππ
14. ππ = π π 36. βππ β π = π βππ
15. ππ = π π 37. βππ β π = π βπ
16. ππ = π π 38. βππ β π = π βππ
17. ππ = βπ βπ 39. π(π + π) = π π
18. βππ = ππ βπ 40. π(π + π) = ππ π
19. βππ = ππ βπ 41. π(π + π) = ππ π
20. βππ = ππ βπ 42. π(π β π) = βπ π
21. βππ = ππ βππ 43. βπ(ππβ π) = βπ π
22. ππ = βππ βπ 44. βπ(ππ+ π) = ππ βπ
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Example 1 (22 minutes)
Review the descriptions of preserves the inequality symbol and reverses the inequality symbol with students.
Example 1
Preserves the inequality symbol: means the inequality symbol stays the same.
Reverses the inequality symbol: means the inequality symbol switches less than with greater than and less than or equal to with greater than or equal to.
Split students into 4 groups. Discuss the directions to the Opening Exercise.
There are four stations. Provide each station with two cubes containing integers. (Cube templates provided at end of document.) At each station, students are to do the following, recording their results in their student materials: (An example is provided for each station.)
1. Roll each die, recording the numbers under the first and third columns. Students are to write an inequality symbol that makes the statement true. Repeat this four times to complete the four rows in the table.
2. Perform the operation indicated at the station (adding or subtracting a number, writing opposites, multiplying or dividing by a number), writing a new inequality statement.
3. Determine if the inequality symbol is preserved or reversed when the operation is performed.
Station #1: Add or Subtract a Number to Both Sides of the Inequality
Station 1
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
βπ < π Add π βπ + π < π + π βπ < π
Preserved
Add βπ
Subtract π
Subtract βπ
Add π
Examine the results. Make a statement about what you notice, and justify it with evidence.
When a number is added or subtracted to both numbers being compared, the symbol stays the same and the inequality symbol is preserved.
Scaffolding:
Guide students in writing a statement using the following: β’ When a number is added or
subtracted to both numbers being compared, the symbol stays the same; therefore, the inequality symbol is preserved.
MP.2 &
MP.4
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Station #2: Multiply each term by β1
Station 2
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
βπ < π Multiply
by βπ
(βπ)(βπ) < (βπ)(π)
π < βπ
π > βπ
Reversed
Multiply by βπ
Multiply by βπ
Multiply
by βπ
Multiply
by βπ
Examine the results. Make a statement about what you notice and justify it with evidence.
When both numbers are multiplied by βπ, the symbol changes and the inequality symbol is reversed.
Station #3: Multiply or Divide Both Sides of the Inequality by a Positive Number
Station 3
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
βπ > βπ Multiply
by ππ
(βπ) οΏ½ πποΏ½ > (βπ) οΏ½
πποΏ½
βπ > βπ Preserved
Multiply
by π
Divide by π
Divide by ππ
Multiply by π
Examine the results. Make a statement about what you notice, and justify it with evidence.
When a positive number is multiplied or divided to both numbers being compared, the symbol stays the same and the inequality symbol is preserved.
Scaffolding:
Guide students in writing a statement using the following: β’ When βπ is multiplied to both
numbers, the symbol changes; therefore, the inequality symbol is reversed.
Scaffolding:
Guide students in writing a statement using the following: When a positive number is
multiplied or divided to both numbers being compared, the symbol stays the same; therefore, the inequality symbol is preserved.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Station #4: Multiply or Divide Both Sides of the Inequality by a Negative Number
Station 4
Die 1 Inequality Die 2 Operation New Inequality Inequality Symbol Preserved or Reversed?
π > βπ Multiply
by βπ π(βπ) > (βπ)(βπ)
βπ < π Reversed
Multiply by βπ
Divide by βπ
Divide by
βππ
Multiply
by βππ
Examine the results. Make a statement about what you notice and justify it with evidence.
When a negative number is multiplied or divided to both numbers being compared, the symbol changes and the inequality symbol is reversed.
Discussion
Summarize the findings and complete the lesson summary in the student materials.
To summarize, when did the inequality change and when did it stay the same?
The inequality reverses when we multiply or divide the expressions on both sides of the inequality by a negative number.
Exercise (5 minutes) Exercise
Complete the following chart using the given inequality, and determine an operation in which the inequality symbol is preserved and an operation in which the inequality symbol is reversed. Explain why this occurs.
Solutions may vary. A sample student response is below.
Inequality Operation and New
Inequality Which Preserves the Inequality Symbol
Operation and New Inequality which Reverses
the Inequality Symbol Explanation
π < π Add 4 to both sides π < π
π+ π < π+ π π < π
Multiply both sides by -4 π < π
π(βπ) > π(βπ) βπ > βππ
Adding a number to both sides of an inequality preserves the inequality symbol. Multiplying a negative number to both sides of an inequality reverses the inequality symbol.
Scaffolding:
Guide students in writing a statement using the following: When a negative number is
multiplied by or divided by a negative number, the symbol changes; therefore, the inequality symbol is reversed.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
βπ > βπ Subtract 3 to both sides βπ > βπ
βπ β π > βπβ π βπ > βπ
Divide both sides by -2 βπ > βπ
βπ Γ·βπ < βπΓ· βπ π < π
Subtracting a number to both sides of an inequality preserves the inequality symbol. Dividing a negative number to both sides of an inequality reverses the inequality symbol.
βπ β€ π Multiply both sides by 3 βπ β€ π
βπ(π) β€ π(π) βπ β€ π
Multiply both sides by -1 βπ β€ π
βπ(βπ) β₯ π(βπ) π β₯ βπ
Multiplying a positive number to both sides of an inequality preserves the inequality symbol. Multiplying a negative number to both sides of an inequality reverses the inequality symbol.
βπ + (βπ)< βπβ π
Add 5 to both sides βπ + (βπ) < βπβ π
βπ + (βπ) + π < βπβ π+ π π < π
Multiply each side by β Β½ βπ + (βπ) < βπ β π
βπ < βπ
βποΏ½βπποΏ½ > βποΏ½β
πποΏ½
ππ
> π
Adding a number to both sides of an inequality preserves the inequality symbol. Multiplying a negative number to both sides of an inequality reverses the inequality symbol.
Closing (3 minutes)
What does it mean for an inequality to be preserved? What does it mean for the inequality to be reversed? When does a greater than become a less than?
Exit Ticket (5 minutes)
Lesson Summary
When both sides of an inequality are added or subtracted by a number, the inequality symbol stays the same and the inequality symbol is said to be preserved.
When both sides of an inequality are multiplied or divided by a positive number, the inequality symbol stays the same and the inequality symbol is said to be preserved.
When both sides of an inequality are multiplied or divided by a negative number, the inequality symbol switches from < to > or from > to <. The inequality symbol is reversed.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Name ___________________________________________________ Date____________________
Lesson 12: Properties of Inequalities
Exit Ticket 1. Given the initial inequality β4 < 7, state possible values for π that would satisfy the following inequalities:
a. π(β4) < π(7)
b. π(β4) > π(7)
c. π(β4) = π(7)
2. Given the initial inequality 2 > β4, identify which operation preserves the inequality symbol and which operation reverses the inequality symbol. Write the new inequality after the operation is performed.
a. Multiply both sides by β2.
b. Add β2 to both sides.
c. Divide both sides by 2.
d. Multiply both sides by β 12.
e. Subtract β3 from both sides.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Exit Ticket Sample Solutions
1. Given the initial inequality βπ < π, state possible values for π that would satisfy the following inequalities:
a. π(βπ) < π(π)
π > π
b. π(βπ) > π(π)
π < π
c. π(βπ) = π(π)
π = π
2. Given the initial inequality π > βπ, identify which operation preserves the inequality symbol and which operation reverses the inequality symbol. Write the new inequality after the operation is performed.
a. Multiply both sides by βπ.
Inequality symbol is reversed.
π > βπ π(βπ) < βπ(βπ)
βπ < π
b. Add βπ to both sides.
Inequality symbol is preserved.
π > βπ π + (βπ) > βπ + (βπ)
π > βπ
c. Divide both sides by π.
Inequality symbol is preserved.
π > βπ π Γ· π > βπ Γ· π
π > βπ
d. Multiply both sides by βππ.
Inequality symbol is reversed.
π > βπ
ποΏ½βπποΏ½ < βποΏ½β
πποΏ½
βπ < π
e. Subtract βπ from both sides.
Inequality symbol is preserved.
π > βπ π β (βπ) > βπ β (βπ)
π > βπ
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Problem Set Sample Solutions
1. For each problem, use the properties of inequalities to write a true inequality statement. Two integers are βπ and βπ.
a. Write a true inequality statement.
βπ < βπ
b. Subtract βπ from each side of the inequality. Write a true inequality statement.
βπ < βπ
c. Multiply each number by βπ. Write a true inequality statement.
ππ > π
2. In science class, Melinda and Owen are experimenting with solids that disintegrate after an initial reaction. Melindaβs sample has a mass of πππ grams, and Owenβs sample has a mass of πππ grams. After one minute, Melindaβs sample lost one gram and Owenβs lost three grams. For each of the next ten minutes, Melindaβs sample lost one gram per minute and Owenβs lost three grams per minute.
a. Write an inequality comparing the two sampleβs masses after one minute.
Melindaβs sampleβs loss: βπ gram
Owenβs sampleβs loss: βπ gram
πππ < πππ
b. Write an inequality comparing the two masses after four minutes.
Melindaβs sampleβs loss after π minutes: (βπ) = βπ
Owenβs sampleβs loss after π minutes: (βπ) = βππ
πππ < πππ
c. Explain why the inequality symbols were preserved.
Neither mass was multiplied or divided by a negative number, so the inequality symbol stayed the same.
3. On a recent vacation to the Caribbean, Kay and Tony wanted to explore the ocean elements. One day they went in a submarine πππ feet below sea level. The second day they went scuba diving ππ feet below sea level.
a. Write an inequality comparing the submarineβs elevation and the scuba diving elevation.
βπππ < βππ
b. If they only were able to go one-fifth of the capable elevations, write a new inequality to show the elevations they actually achieved.
βππ < βππ
c. Was the inequality symbol preserved or reversed? Explain.
The inequality symbol was preserved because the number that was multiplied to both sides was NOT negative.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
4. If π is a negative integer, then which of the number sentences below is true? If the number sentence is not true, give a reason.
a. π + π < π
True
b. π + π > π
False because adding a negative number to π will decrease π which will not be greater than π.
c. π β π > π
True
d. π β π < π
False because subtracting a negative number is adding a number to π which will be larger than π.
e. ππ < π
True
f. ππ > π
False because a negative number is being multiplied.
g. π + π > π
True
h. π + π < π
False because adding π to a negative number is greater than the negative number itself.
i. π β π > π
True
j. π β π < π
False because subtracting a negative number is the same as adding the number, which is greater than the negative number itself.
k. ππ > π
False because a negative number is being multiplied.
l. ππ < π
True
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 12
Die Templates: