Module 2: Compound Inequalities
Section II: Functions, Inequalities, and the Absolute Value
[pic]
Module 2: Compound Inequalities
[pic] example: [pic] is a compound inequality since it can be rewritten as two inequalities:
[pic]
| |
|To solve a compound inequality, rewrite it as two inequalities and solve each of these inequalities. (See the Section I: Review if you |
|don’t remember how to solve an inequality.) |
[pic] example: Solve [pic]. Write the solution in interval notation and graph the solution on a number line.
SOLUTION:
[pic]
We can combine [pic] and [pic] into the compound inequality [pic]. So, the solution to [pic] is the set [pic], which we can write in interval notation: [pic]. Below is a graph of the solution on a number line:
[pic]
| |
|Remember that on a graph, an empty circle indicates that the point is not included and a solid circle indicates that the point is included. |
|(So in the graph above, –1 is not included while 2 is included.) |
[pic] example: Solve the compound inequality [pic]. Write the solution in interval notation and graph the solution on a number line.
SOLUTION: Here, the compound inequality is already written in two parts. We’ll solve each part independently:
[pic]
Thus, the solution to [pic] is [pic]; a graph of the solution is given below:
[pic]
[pic]
|[pic] |CLICK HERE FOR A COMPOUND INEQUALITY EXAMPLE |
[pic]
|[pic] |CLICK HERE FOR ANOTHER COMPOUND INEQUALITY EXAMPLE |
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|[pic] |CLICK HERE FOR SOME INEQUALITY CAUTIONS |
[pic]
[pic] Try these yourself and check your answers.
a. Solve [pic]. Write your solution in interval notation.
b. Solve [pic]. Write your solution in interval notation.
SOLUTIONS:
a.
[pic]
So the solution to [pic] is [pic].
b.
[pic]
The solution to [pic] is [pic].
[pic]
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