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]

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|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]

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|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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