Download Solving Unit Inequalities B ACTIVIES

Activity #1: Inequalities – Representation using Number Line
How to use a Number Line to represent an inequality:
The symbol > means “is greater than.”
The symbol < means “is less than.”
The symbol ≥ means “is greater than or equal to.”
The symbol ≤ means “is less than or equal to.”y
To represent all the integers greater than 5, you can write {6, 7, 8, 9, …}
or write x > 5 where x is an integer. You can also use a number line.
–1
0
+1 +2 +3 +4 +5 +6 +7 +8 +9
To represent all the integers greater than or equal to 5, you can write
{5, 6, 7, 8, 9,…}, write x ≥ 5 where x is an integer, or use a number line.
–1
0
+1 +2 +3 +4 +5 +6 +7 +8 +9
Activity #2: Inequalities – Yukon Quest
The Yukon Quest is an International Dog Sled Race held each year over February. No
more than 50 dog teams are allowed in this 1600 km race through the Northern
wilderness between Whitehorse, Yukon and Fairbanks, Alaska. The starting points
alternate between the two cities – even years from Fairbanks and odd years from
Whitehorse. The race starts on schedule regardless of temperature and weather
conditions. The number of entrants can be 1, 2, 3, … 50.
To enter the Yukon Quest a driver must be at least 18 years old.
a) Mira is 21 years old. Write an inequality to compare Mira’s age with the age
restriction.
b) Lucas is 17 years old. Write an inequality to compare Mira’s age with the age
restriction.
c) The symbol > means “is greater than or equal to”. Use this symbol to model the
statement “a sled driver must be at least 18 years of age”.
d) To what subset of the real numbers do the possible ages belong?
The optimal outdoor temperature for a dog to run is above -10 oC but not above
2 oC . This means its paws will neither freeze nor blister from overheating. The
collection of all optimal temperatures for a dog to run is a set.
a) List 10 possible temperatures for a dog run.
b) To what set of real numbers do these belong? Why would it NOT be
appropriate to use the set of integers in this situation?
c) Let t represent the set of temperatures. Write an inequality to represent
the temperatures above -10 oC.
d) Let t represent the set of temperatures. Write an inequality to represent
the temperatures not above 2 oC.
e) Represent c & d above as a single inequality.
f) Draw a number line to represent the optimal temperatures for a dog to
run. How can you show that 10 oC is not a possibility but 2 oC is?
g) What are THREE possible ways to represent a set of numbers?
Activity #3: Inequalities – Set Notation & Graphic Representations
A skating party is planned for the community lake. In order for the event to take place
the temperature must be above -5 oC but not above 6 oC.
We can represent this set of temperatures using two inequalities. Let y represent all
real numbers in the set.
The temperature must be above -5 oC: y > -5 oC
The temperature must be 6 oC or less: y < 6 oC
Example: Describe these sets in words and write each in set notation.
a) The set of Natural Numbers greater than or equal to 2.
Set notation will be { x  x > 2, x  N}
b) The set of Real Numbers greater than or equal to -2 and less than but not equal
to 2.
Set notation will be { x  -2 < x < 2, x  R}
Practice:
Mathematics 9 Textbook p. 148-149 - Check Your Understanding #1-17
Choose samples from each question on these pages as practice.
Activity #4: Inequalities – Practice – Set Notation & Graphic Representations
1. Write the set notation for each number line representation.
a)
________________________________
b)
________________________________
c)
________________________________
d)
________________________________
e)
________________________________
f)
________________________________
2. Describe a situation that could make sense for each of the sets described in question 1 above
.
a) ________________________________
b) ________________________________
c) ________________________________
d) ________________________________
e) ________________________________
f) ________________________________
3. Describe each of the following set notations in words.
a) {x | –3 ≤ x ≤ 5, x  W}
________________________________
________________________________
b) {n | n < –1, n  W}
________________________________
________________________________
c) {s | s > 10, s  R}
________________________________
________________________________
d) {m | m < 0, m  R}
________________________________
________________________________
e) {p | p ≥ 1, p  N}
________________________________
________________________________
f) {y | 1 ≤ y < 8, y  N}
________________________________
________________________________
4. Represent each of the inequalities from question 3 on a number line.
5. Express each of the following using set notation :
a) All real numbers greater than 11 but less than 20
________________________________
b) All whole numbers smaller than – 4
________________________________
c) All natural numbers less than or equal to 5
________________________________
d) All real numbers greater than or equal to –2 but less than 9
________________________________
e) All positive natural numbers smaller than 6
________________________________
f) All whole numbers less than –7 but greater than or equal to –15
________________________________
g) All natural numbers greater than 2
________________________________
6. Explain why {x | x < –3, x  W} and {x | x < –3, x  R} are different. Use a number line to help
clarify your explanation.
7. Represent this set using set notation and in words.
___________________________________
___________________________________
___________________________________
___________________________________
8. An airplane can transport 100 people, including 5 crew members. Represent this information in
set notation.
___________________________________
9. A daycare accepts children between and including the ages of 4 and 9 years.
a) Represent this using set notation.
________________________________
b) Represent this using a number line.
10. Minimum speed on a NS highway is 80 km/h and maximum speed is 110 km/h. Represent this
using set notation.
___________________________________
___________________________________
Activity #5: Exploration - Inequalities
Complete the chart below for each inequality to investigate how the operation affects
the inequality.
Decide whether or not the resulting inequality is still true (T) or if it is False (F).
Use the inequality: – 2 ˂ 4 with EACH operation.
Number
Use this inequality
each time
Add a positive
number to both
sides
Subtract a positive
Still True (T) or Is it
Now False (F)
–2 ˂ 4
Add +3
–2+3˂ 4+3
1<7
T
(because 1 is less
than 7)
Multiply by a
positive
Divide by a
positive
Add a negative
Subtract a
negative
Multiply by a
negative
Divide by a
negative
When is the inequality NOT true? What adjustment can you make so that it will be
true?
This time use the inequality 5 ˃ 1 each time.
Add a positive
number
Subtract a positive
Number
3
5 +3 ˃ 1+3
8˃4
True or False
T
Multiply by a
positive
Divide by a
positive
Add a negative
Subtract a
negative
Multiply by a
negative
Divide by a
negative
When is the inequality NOT true? What adjustment can you make so that it will be
true?
Activity #6: Solving Inequalities - Lesson
Solving inequalities looks very much like solving equations with a few differences.
A. View ShowMe video “Solving Inequalities” OR study the examples shown here:
Example #1:
2y + 3 > 7
2y + 3 – 3 > 7 – 3
2y > 4
2y > 4
2 2
y>2
Example #2:
Activity #7: Solving Inequalities - Practice
Solve the following inequalities using EITHER symbols or balances. Check your work.
Inequality
Check
Solve 8 of the following inequalities. Check your work.
Inequality
8  3x  7 x  12
Check
Inequality
 3(2 x  5)  0,6(6 x  7)
7x < 2x + 30
1
1
(4 x  8)  x  8
4
2
2 y  7 y  15
3
1
y4 y6
4
4
5x  4  2 x  5
3y 5  y 3
 

5 2
5 2
8( x  7)  2 x  1
Find values for x that give an
c
> 3.2
5
area less than 40 cm
2
in the rectangle to
the right.
Find values for m that give a
rectangle to the right.
perimeter greater than or equal to 106 m in the
Check
Activity #8: Solving Inequalities – Modeling using Number Line and Set Notation
1. Solve each inequality. Represent each solution on a number line. Assume x  R.
a) x – 2 < 5
b) –4x > 12
c) 3x + 2 ≥ 5
d)
> –2
e) –3x + 8 ≤ 17
f) 3(x – 1) < 9
g) 2x + 4 ≥ 5x + 19
2. Which inequality from question 1 has a solution that could also be from natural numbers?
___________________________________
___________________________________
3. Represent the solution from each inequality in question 1 using set notation. Assume x  Z or R .
4. Represent the solution for each of the following inequalities in set notation.
a) x + 8 > 3x – 4, where x  Z
________________________________
b) 5y – 9y ≥ 14, where y  R
________________________________
c) 4(m – 2) ≤ 24, where m  N*
________________________________
d)
> 4, where p  Z
________________________________
e) 0,6x + 2 ≥ 5,6, where x  R
________________________________
f) 22 > 3n + 2n, where n  R
________________________________
g) 20 ≤ 2(3k + 1), where k  N
________________________________
5. Represent each of the following using set notation.
a)
________________________________
b)
________________________________
c)
________________________________
d)
________________________________
e)
________________________________
f)
Activity #9: Final Review - Solving Equations and Inequalities
Solve as many of the following equations and inequalities as possible. Show all your work. Check
your answers with a calculator.
18. d – 7 > –10
1.
2.8(3d – 2) = –12
2.
3(d + 0.4) = –3.9
19.
3.
y
1
+1=
2
4
20. 5x + 4 ≤ 6x – 1
4.
2
3
1
y+
=– y
3
2
12
21.
5.
0.45 – 0.3g = 0.85 + 0.2g
6.
1
3
1
– x=–
3 2
6
7.
2x – 22 = 2.5x + 18
8.
3.1(2x – 1.5) = – 6
9.
4.8 + 1.1f = 1.5
10. 3(u – 12.5) = –3.41
c
> 3.2
5
x
– 5 < 10
3
22. –2(3x + 4) ≥ – 8
23. 5 – x < 2
24. 12 – 8x < 17 – 3x
25.
x
+6>8
3
11. 3x – 2 = 5
26. 7x < 2x + 30
12. 0.25r – 0.32 = 0.45r
13. – 0.25f = 0.35 – 0.45f
27.
y
 2 < 16
6
14. 4(4 – n) = 6n – 4
15.
2
2
r
= –
5
5
3
16. 2.05 = 0.9x – 6.5
17. 2.05 = 0.9x – 6.5
28. –5y + 92 ≥ 40
29. 8( x  7)  2 x  1
Solve as many of the following problems algebraically as possible. Show all your work. Check that
your answers make sense.
30. During basketball seasons over the last number of years, MSMS won 4 games more than FBJH, and
Brookside won 13 games more than FBJH. Together, the three schools won 80 games. How many games
did each team win?
31. A rectangular garden has a perimeter of 66 m. Its length is triple its width plus 1 m. FInd the exact
dimensions of the garden.
32. During hockey season, Michaud scored twice as many goals as Simard, and Fraser scored 10 goals
more than Michaud. If they scored 260 goals altogether, how many goals did each player score ?
33. Find the values for x that give an area less than 80 cm2 in this rectangle :
34. FInd the value of m if the perimeter of this rectangle is 212 m.
35.
The square and equilateral triangle below have the SAME perimeter. Find the length of each side.
36. These 2 rectangles have the SAME area. Find the length of each side.