Notes for Lesson 3-6: Compound Inequalities
3-6.1 - Solving Compound Inequalities
The inequalities we have seen so far are simple inequalities. When two simple inequalities are combined into
one statement by the words AND or OR the result is called a compound inequality.
Vocabulary:
Compound Inequality - two inequalities that are combined into one statement by the words and or or.
Example:
A water analysis recommends that the pH level of a swimming pool be between 7.2 and 7.6 inclusively. Write
a compound inequality to show the pH levels that are within the recommended range.
pH  7.2 and pH  7.6
7.2  pH  7.6
3-6.2 - Solving compound inequalities involving AND
To solve a compound inequality, separate the two inequalities and solve both separately.
Examples: Solve and graph each compound inequality.
5  2 x  3  9
4 x28
5  2 x  3 AND 2 x  3  9
4  x  2 AND x  2  8
8  2 x AND 2 x  6
2  x AND x  6
4  x AND x  3
x
x
-8 -6 -4 -2
0
2
4
6
-8 -6 -4 -2
8
0
2
4
6
8
Example: −3 ≤ 𝑚 − 4 < −1
3-6.3 - Solving compound inequalities involving OR
Examples: Solve and Graph each compound inequality
4  a  1 OR  4  a  3
2 x  6 OR 3 x  12
a  5 OR a  1
x  3 OR x  4
x
-8 -6 -4 -2
0
2
4
6
8
Example: 3𝑥 + 2 < −7 𝑜𝑟 − 4𝑥 + 5 < 1
x
-8 -6 -4 -2
0
2
4
6
8
3-6.4 - Writing a compound inequality from a graph
x
-8 -6 -4 -2
0
2
4
6
x
8
The shaded portion of the graph is not
between the two values so the compound
inequality involves OR
-8 -6 -4 -2
0
2
4
6
8
The shaded portion of the graph is between the
the two values, so the compound inequality involves
AND
x  0 AND x  6
x  1 OR x  7
or
0 x6
3-6.5 – Using Interval Notation
Vocabulary:
Interval Notation – A notation for describing an interval on a number line
You can use interval notation to show an inequality. Interval notation uses 3 symbols. Brackets, Parentheses
and the infinity sign. You use the parentheses to show > or <, a bracket to show ≥ 𝑜𝑟 ≤ and the infinity sign
∞ to show when it goes on forever in a direction
Example:
𝑥 ≥ 2 [2, ∞) The bracket shows to include the 2 and the ) to show it goes on without an endpoint
𝑥 < 2 (−∞, 2) The ( to show it goes on forever and ) shows the 2 is not included
−4 ≤ 𝑥 < 6 [−4, 6) Include the – 4 on the low side and do not include the 6 on the high side
𝑥 ≤ −1 𝑜𝑟 𝑥 > 2 (−∞, −1] 𝑜𝑟 (2, ∞)
How do you write (−2, 7] as an inequality
How do you write (7, ∞) as an inequality
−2 < 𝑥 ≤ 7 or 𝑥 > −2 𝑎𝑛𝑑 𝑥 ≤ 7
𝑥>7