Inequality: -5 < x < -2

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Section 2.9
Linear Inequalities, Part 2
Compound Inequalities
A compound inequality contains two
inequality symbols.
Example: 0  4(5 – x) < 8
This means that 0  4(5 – x) and
4(5 – x) < 8 must both be true.
Interval Notation for Compound Inequalities:
• Inequality: -5 < x < -2
– The interval notation (-5,-2) represents all the numbers in
between -2 and -5, excluding -2 and -5.
• Inequality: -5 < x ≤ -2
– The interval notation (-5,-2] represents all the numbers in
between -2 and -5, including -2 and excluding -5.
• Inequality: -5 ≤ x < -2
– The interval notation [-5,-2) represents all the numbers in
between -2 and -5, excluding -2 and including -5.
• Inequality: -5 ≤ x ≤ -2
– The interval notation [-5,-2] represents all the numbers in
between -2 and -5, including -2 and -5.
Example from today’s homework:
(  7,1)
Example
Graph:  2  x  5
How would you write this in interval notation?
Answer: (-2, 5]
To solve a compound inequality, perform operations
simultaneously to all three parts of the inequality (left, middle, and
right) until you get the variable isolated by itself in the middle.
Example: Solve the inequality 9 < z + 5 < 13 , then graph the
solution set and write it in interval notation.
9 < z + 5 < 13
9 – 5 < z + 5 – 5 < 13 – 5
4<
z
<
8
Graph:
Interval notation:
(4, 8)
Subtract 5 from all three parts.
Example:
Solve the inequality 0  4(5 – x) < 8 . Graph the
solution set and write it in interval notation.
0  20 – 4x < 8
0 – 20  20 – 20 – 4x < 8 – 20
Use the distributive property.
Subtract 20 from each part.
– 20  – 4x < – 12
Simplify each part.
5 x >3
Divide each part by –4.
3< x 5
Reverse to put in standard form.
Remember that the sign changes direction when you divide
by a negative number.
Graph:
Interval notation: (3,5]
Example from today’s homework:
Example
You are having a catered event. You can spend at
most $1200. The set up fee is $250 plus $15 per
person, find the greatest number of people that can
be invited and still stay within the budget.
Let x represent the number of people
Set up fee + cost per person × number of people ≤ 1200
250 +
15x
≤ 1200
Example continued:
You are having a catered event. You can spend at most $1200. The
set up fee is $250 plus $15 per person, find the greatest number of
people that can be invited and still stay within the budget.
250  15 x  1200
15 x  950
15 x
15

950
15
x  6 3 .3
The number of people who can be invited must be 63
or less to stay within the budget.
The assignment on this material (HW 10) is due at the
start of the next class session.
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