Download MAT 1033 - Compound Inequalities and Interval Notation

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Compound Inequalities and Interval Notation
Dr. Gisela Acosta-Carr.
A compound inequality contains at least two inequality symbols. The following are some
examples of compound inequalities:
a. -3 ≤
≤ 3 states that x is any number between -3, and 3, including -3 and 3.
b. 0 <
< 5 states that x is any number between 0 and 5, without including the endpoints.
c. x < -3 or x ≥ 2 states that the values of x are either less than -3 or greater than or equal to 2.
We can express the solution set to a compound inequality by graphing on a number line, or
using interval notation. The graphical solution sets of the previous compound inequalities are
shown below.
a. -3 ≤
≤3
-5
-3
-4
-2
-1
1
0
2
3
4
5
We can also use brackets to graph the solution set.
-5
-4
[-3
b. 0 <
<5
-4
-2
-3
-2
-1
-1
1
0
1
0
2
]3
2
3
4
4
5
5
6
We can also use parentheses to graph the solution set.
-4
-3
-2
-1
(0
1
2
3
)5
4
6
c. x < -3 or x ≥ 2
-5
-4
)-3
-2
-1
0
1
[2
3
4
5
Since we can express the solution set to a compound inequality by graphing on a number line,
or using interval notation, let us now review interval notation.
Interval notation is frequently used to express a set of numbers between two values, a and b.
We basically use two symbols: parentheses ( ) and brackets [ ]:
( ) is used when we have the inequality symbols less than, < , or greater than, >. This means
that specified values for a or b are not included.
[ ] is used when we have the inequality symbols less than or equal to, ≤, or greater than or
equal to, ≥. This means that specified values for a or b are included.
Page 1 of 7
Example 1
The inequality -3 < < 3 reflects all the real numbers between -3 and 3, without including -3
nor 3. Using interval notation, we write this inequality as (-3, 3).
Example 2
The inequality -3 ≤ ≤ 3 reflects all the real numbers between -3 and 3, including -3 and 3.
Using interval notation, we write this inequality as [-3, 3].
Example 3
The inequality -3 ≤ < 3 reflects all the real numbers between -3 and 3, including -3 but not 3.
In interval notation, [-3, 3).
Example 4
The inequality -3 < ≤ 3 reflects all the real numbers between -3 and 3, including 3 but not -3.
In interval notation, (-3, 3].
Interval Notation
A closed interval [a, b] describes all real numbers x, such that a ≤ x ≤ b
An open interval (a, b) describes all real numbers x, such that a < x < b
A half-open interval (also called half-closed) describes one of the following:
[a, b) describes all real numbers x, such that a ≤ x < b
(a, b] describes all real numbers x, such that a < x ≤ b
Caution: When using interval notation, be careful to always write the smaller value to the left
and the greater value to the right.
Interval Notation with Infinity Symbol
The symbol "∞" is called the infinity symbol and we use it when there is no lower or upper
bound on the number line. For example, we know that the inequality ≥ 3 includes all real
numbers greater than or equal to 3, without limit. The corresponding graph is:
-3
-2
-1
0
1
2
[3
4
5
There is no upper bound, thus the interval is unlimited, and the interval notation is [3, ∞).
Infinite Intervals
Use ∞ to signify that the values continue getting larger without end (unbounded to the right on
the number line). Use -∞ to signify that the values continue getting smaller without end
(unbounded to the left on the number line).
[a, ∞) describes all real numbers x, such that x ≥ a
(a, ∞) describes all real numbers x, such that x > a
(-∞, a] describes all real numbers x, such that x ≤ a
(-∞, a) describes all real numbers x, such that x < a
Page 2 of 7
Important: Since the use of the infinity symbol implies that the interval is unlimited, we never
close it with a bracket. Always use a parenthesis next to the ∞ symbol!
Example 5
The inequality
-4
-3
-2
< 3 reflects all the real numbers less than 3.
-1
0
1
2
)3
4
5
In interval notation, we write this inequality as (-∞, 3).
Example 6
Use inequality and interval notation to write the set of numbers that are:
a. between -7 and 9.
b. between -5 and 4.4, including -5 but excluding 4.4.
c. greater than or equal to 6.
d. less than -12.
e. between 0 and 3/4, including the endpoints.
f. all the real numbers.
Inequality
Notation
a. -7 < x < 9
Interval Notation
(-7, 9)
[6, ∞)
Half-open or
Half-closed
Infinite
(-∞, -12)
Infinite
e. 0 ≤ x ≤ 3/4
[0, 3/4]
Closed
f. -∞ < x < ∞
(-∞, ∞)
Infinite
b. -5 ≤ x < 4.4
c. x ≥ 6
d. x < -12
[-5, 4.4)
Type of
Interval
Open
More Compound Inequalities
1 ≤ x + 4 ≤ 7 states that x is any number that is a solution to both inequalities x + 4 ≥ 1 and
x + 4 ≤ 7. For example, x = 2 is a solution to the given compound inequality because it will
satisfy both individual simple inequalities: 2 + 4 ≥ 1 and 2 + 4 ≤ 7.
x + 5 < 1 or x + 3 > 8 means that x is any number that is a solution to either inequality. For
example, x = -7 is a solution to the compound inequality because it satisfies x + 5 < 1. The value
x = 6 is also another solution to the compound inequality since it satisfies x + 3 > 8.
Example 6
Solve the inequality 2 ≤ x + 3 ≤ 7.
Observe that this is equivalent to saying x + 3 ≥ 2 and x + 3 ≤ 7.
Page 3 of 7
We can solve each individual inequality:
x+3≥2
and
x+3≤7
x+3−3≥2−3
x+3−3 ≤7−3
x ≥ -1
x≤4
Note: Review "Solving Inequalities" in Section 2.4.
Graphically, we can see that the solution set to the given compound inequality contains all the
real numbers between -1 and 4, including the endpoints.
-5
-4
-3
-2
[
-1
0
1
2
3
]4
5
The solution set using interval notation is [-1. 4].
Direct approach: We can solve the given inequality applying a direct approach -all at once- as
follows.
2≤x+3≤7
2 − 3 ≤ x + 3 − 3 ≤ 7− 3
To isolate x, we subtract 3 from all three parts of the inequality.
-1 ≤ x ≤ 4
In interval notation, [-1. 4].
Example 7
Solve the inequality 19 ≤ 5x + 4 < 44. Write your answer using interval notation.
We can solve each individual inequality:
5x + 4 ≥ 19
and
5x + 4 < 44
5x + 4 − 4 ≥ 19 − 4
5x + 4 − 4 < 44 − 4
5x ≥ 15
5x < 40
x ≥3
x<8
The solution set using interval notation is [3, 8).
Let us solve the given inequality by applying a direct approach (all at once):
19 ≤ 5x + 4 < 44
19 − 4 ≤ 5x + 4 − 4 < 44 − 4
Make sure you subtract 4 from all three parts.
15 ≤ 5x < 40
≤
<
Make sure you divide all three parts by 5.
3≤x<8
In interval notation, [3, 8).
Page 4 of 7
Example 8
Solve the inequality -8 < -2x < 12. Write your answer using interval notation.
Solving each individual inequality, we have:
-2x > -8
and
-2x < 12
<
>
x<4
Remember to reverse the inequality symbol when dividing by a negative.
x > -6
The solution set using interval notation is (-6, 4).
Solving by applying a direct approach:
-8 < -2x < 12
>
>
Make sure to reverse all inequality symbol when dividing by a negative.
4 > x > -6
This is equivalent to x > -6 and x < 4.
In interval notation, (-6, 4).
Example 9
Solve the inequality -29 < 3(x − 9) ≤ 201. Write your answer using interval notation.
We will apply a direct approach:
-29 < 3x − 27 ≤ 201
Simplify by applying the distributive property.
-29 + 27 < 3x − 27 + 27 ≤ 201 + 27
Add 27 to all three parts.
-2 < 3x ≤ 228
<
≤
Divide all three parts by 3.
The solution set using interval notation is (-2/3, 76].
Example 10
Solve x + 2 < -1 or 6x > 24. Write your answer using interval notation.
x + 2 < -1 or 6x > 24 means that x is any number that is a solution to either individual
inequality. Since we have a compound inequality involving "or," we will solve each inequality:
x + 2 < -1
or
6x > 24
>
x + 2 − 2 < -1 − 2
x < -3
x>4
Graphically, we can see that the solution set to the given compound inequality contains all the
real numbers less than -3, or greater than 4:
)
-5 -4 -3 -2 -1
0 1
2
3
(4
5
6
7
The solution to the compound inequality will be the union of the two solution sets, which in
interval notation we write as (-∞, -3) ∪ (4, ∞).
Page 5 of 7
Practice Problems
Note: You will not turn in these problems; they will help you practice solving compound
inequalities and using interval notation. The answer key is included here so that you can check
your work. See the answer key starting on page 7.
1. Use inequality, graphical, and interval notation on the table that follows to write
the set of numbers that are:
a. between -5 and 6.
b. less than 1.5.
c. greater than or equal to -5.
d. between -4 and 0, including the endpoints.
e. between -3.5 and 2, including -3.5 but excluding 2.
Inequality
Notation
Graphical Notation
Interval Notation
Type of
Interval
a.
b.
c.
d.
e.
2. Write the given graphical intervals in interval notation.
a.
b.
-5
(
-4
-3
-2
-1
0
1
2
3
]5
4
c.
-5
-4
-3
-2
-1
-5
-4
-3
-2
-1
-5
-4
-3
-2
0
(
1
2
3
)4
0
1
2
3
4
5
d.
[
-5
-4
-3
-2
-1
]0
1
2
3
4
5
e.
[
5
f.
-5
-4
-3
-2
-1
0
1
)2
3
4
5
-1
0
1
[2
3
4
5
3. Write the given intervals using inequality notation.
a. [-5, 8)
b. (1, 7)
c. (0, 9.5]
d. (-∞, 30)
e. [0, 18.3]
f. [1/2, ∞)
4. Missy was asked to write x > -9 in interval notation. Her answer was (-9, ∞] but her
instructor marked it wrong. Explain why.
5. Solve the following inequalities. Write your answers using interval notation.
a. -6 ≤ x + 9 ≤ 11
b. 242 ≤ 10x + 2 < 302
d. 0 < 7(x − 2) ≤ 49
e. 5x < -30 or x − 2 ≥ 10
Page 6 of 7
c. -1 < -2x < 4
Answer Key
1.
Inequality
Notation
Graphical Notation
(
a. -5 < x < 6
-5
-4
-3
-2
-1
1
0
2
b. x < 1.5
-4
c. x ≥ -5
-3
-1
1
0
4
)2
3
4
-3
-2
-1
0
1
2
3
4
-5
-4
[
-3
-2
-1
]0
1
2
3
4
5
-5
-4
[ -3
-2
-1
0
1
)2
3
4
5
b. (0, 4)
3. a. -5 ≤ x < 8
d. x < 30
c. [-5, 0]
(-5, 6)
Open
(-∞, -1.5)
Infinite
[-5, ∞)
Infinite
[-4, 0]
Closed
[-3.5, 2)
Half-open or
Half-closed
5
-4
e. -3.5 ≤ x < 2
Type of
Interval
)6
5
[-5
d. -4 ≤ x ≤ 0
2. a. (-4, 5]
-2
3
Interval
Notation
5
x
d. [-1, ∞)
e. (-∞, 2)
b. 1 < x < 7
c. 0 < x ≤ 9.5
e. 0 ≤ x ≤ 18.3
f. x ≥ 1/2
f. [2, ∞)
4. Missy incorrectly used a bracket, closing the infinity symbol. She had to use a parenthesis
following the ∞ symbol: (-9, ∞)
5. a. [-15, 2]
b. [24, 30)
c. (-2, 1/2)
d. (2, 9]
Page 7 of 7
e. (-∞, -6) ∪ [12, ∞).