comparison statements, inequalities and intervals

1. 0
+
– x is negative x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
0
+
–
x is non–positive
x is non–negative
“Non–Positive” vs. “Non–Negative”
List of comparative statements as inequalities:
“More/greater than” vs “Less/smaller than”
C
x is less than C
x is more than C

2. Comparison Statements, Inequalities and Intervals
“No more/greater than” vs “No less/smaller than”
+
–
x is no more than C x is no less than C
C
and “At most” vs “At least”
x is at most C x is at least C
“More than a but no more than b” is “a < x ≤ b”.
+
–
a a < x ≤ b b
Intervals: Let a, b be two numbers such that a < b,
or a ≤ x ≤ b as [a, b],
b
a
or a < x < b as (a, b),
b
a
or a ≤ x < b as [a, b),
b
a
b
a or a < x ≤ b as (a, b],

3. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
∞
a
or a < x, as (a, ∞),
Comparison Statements, Inequalities and Intervals
–∞ a
or x ≤ a, as (–∞, a],
–∞ a
or x < a, as (–∞, a),
Example A. Given intervals I, J, and K
0
–1
–4
J: x > –2 K: –3 < x ≤ 1
I:
a. K U J
–2
–3 1
0
K
J
–3
0
We have
so K U is
We have 1
0
b. K ∩ I –1
–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1
–3
or (–3, ∞).

4. I. (One Piece | |–Inequalities)
If |x| < c then –c < x < c.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 less than 3
–1 5
2
x x
right 3
left 3
Draw
Example C. Translate the meaning of |x – 2| < 3 and solve.
The geometric meaning of the inequality is that “the distance
between x and 0 is 7 or more”. In picture
Example D. Translate the meaning of |x| ≥ 7 and draw.
x < –7 or 7 < x
-7
-7 7
0
x x
I. (Two Piece | |–Inequalities)
If |x| > c then x < –c or c < x.
end point
included

5. Exercise. A. Draw the following Inequalities. Translate each
inequality into an English phrase. (There might be more than
one way to do it)
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
Exercise. B. Translate each English phrase into an inequality.
Draw the Inequalities.
Let P be the number of people on a bus.
1. There were at least 50 people on the bus.
2. There were no more than 50 people on the bus.
3. There were less than 30 people on the bus.
4. There were no less than 28 people on the bus.
Let T be temperature outside.
5. The temperature is no more than –2o.
6. The temperature is at least 35o.
7. The temperature is positive.
Inequalities

6. Inequalities
Let M be the amount of money I have.
8. I have at most $25.
9. I have a non–positive amount of money.
10. I have less than $45.
11. I have at least $250.
Let the basement floor number be given as a negative number
and let F be the floor number that we are on.
12. We are below the 7th floor.
12. We are above the first floor.
13. We are not below the 3rd floor basement.
14. Our floor is at least the 45th floor.
15. We are between the 4th floor basement and the 10th floor.
16. We are in the basement.

7. C. Translate and solve the expressions geometrically.
Draw the solution.
1. |x| < 2 2. |x| < 5 3. |–x| < 2 4. |–x| ≤ 5
5. |x| ≥ –2 6. |–2x| < 6 7. |–3x| ≥ 6 8. |–x| ≥ –5
9. |3 – x| ≥ –5 10. |3 + x| ≤ 7 11. |x – 9| < 5
12. |5 – x| < 5 13. |4 + x| ≥ 9 14. |2x + 1| ≥ 3
Absolute Value Inequalities
15. |x – 2| < 1 16. |3 – x| ≤ 5 17. |x – 5| < 5
18. |7 – x| < 3 19. |8 + x| < 9 20. |x + 1| < 3

8. D. Let I, J, and K be the following intervals:
1. K U I
0
1
–5 J: x < –1 K: –3 < x ≤ 3
I:
9. (K ∩ J) U I
11. Is (K U J) U I = K U (J U I)?
Is (K ∩ J) ∩ I = K ∩ (J ∩ I)?
Is (K ∩ J) U I = K ∩ (J U I)?
Comparison Statements, Inequalities and Intervals
Draw the following intervals and write the answers
in the interval notation.
2. K U J 3. J U I 4. (K U J) U I
5. K ∩ I 6. K ∩ J 7. J ∩ I 8. (K ∩ J) ∩ I
10. K ∩ (J U I)
12. Apu works from 2 pm to before 10 pm,
Bobo works from after 4 pm to exactly midnight.
a. When are they both working?
b. When is that at least one of them is working?

9. E. Express the following intervals as absolute value inequalities
in x.
1. [–5, 5]
Absolute Value Inequalities
2. (–5, 5) 3. (–5, 2) 4. [–2, 5]
5. [7, 17] 6. (–49, 84) 7. (–11.8, –1.6) 8. [–1.2, 5.6]
–2 4
–15 –2
8 38
0 a
–a –a/2
a – b a + b
10.
13.
11.
12.
9.
14.

12. (Answers to odd problems) Exercise A.
1. My nephew is less than 3 years old
3. Today the temperature is less than –8o
-8
x
3
x
3 x
5. Your child must be at least 3 years old to enter the kinder
garden
7. The temperature of the fridge must be at least -8o or else
your food won’t last
-8 x
Inequalities

13. Exercise B.
1. 𝑃 ≥ 50
3. 𝑃 < 30
50 P
30
P
5. 𝑇 ≤ – 2
-2
T
7. 𝑇 > 0
0 T
Inequalities

14. 9. 𝑀 ≤ 0
0
M
11. 𝑀 ≥ 250
250 M
13. 𝐹 > 1
1 F
15. −4 < 𝐹 < 10
-4 F 10
Inequalities

15. Exercise C.
1. [−5, 3] 3. (−∞, 1)
3
–5 1
5. (−3, 1)
–3 -1
7. [−5, 1)
–5 -1
9. [−5, 1)
–5 -1
11. Is (𝐾 ∪ 𝐽) ∪ 𝐼 = 𝐾 ∪ (𝐽 ∪ 𝐼)? Yes
Is (𝐾 ∩ 𝐽) ∩ 𝐼 = 𝐾 ∩ (𝐽 ∩ 𝐼)? Yes
Is (𝐾 ∩ 𝐽) ∪ 𝐼 = 𝐾 ∩ (𝐽 ∪ 𝐼)? No
𝐾 ∩ 𝐽 ∪ 𝐼 = [−5,1)
and 𝐾 ∩ 𝐽 ∪ 𝐼 = (−3,1)
Comparison Statements, Inequalities and Intervals

16. Exercise D.
1. |𝑥| < 2
-2 2
x x
0
3. |– 𝑥| < 2
-2 2
x x
0
5. |𝑥| ≥ – 2
x x
0
7. |– 3𝑥| ≥ 6
-2 2
x x
0
Absolute Value Inequalities

17. 9. |3 – 𝑥| ≥ – 5
x x
0
11. |𝑥 − 9| < 5
4 14
x x
9
13. |4 + 𝑥| ≥ 9
-13 5
x x
-4
15. |𝑥 – 2| < 1
1 3
x x
2
Absolute Value Inequalities

18. 17. |𝑥 − 5| < 5
0 10
x x
5
19. |8 + 𝑥| < 9
-17 -1
x x
-8
Exercise E.
1. |𝑥| ≤ 5 3. 𝑥 < 5 5. 𝑥 − 12 ≤ 5
7. |𝑥 + 6.7| < 5.1 9. |𝑥 − 1| ≥ 3 11. 𝑥 + 8.5 > 6.5
13.|𝑥 +
3𝑎
4
| ≥
𝑎
4
Absolute Value Inequalities