Teaching Notes 3.13: Rewriting Combined Inequalities
as One Inequality
When students solve combined inequalities, they must express their solution as one inequality.
Many, however, have difficulty combining two inequalities as one.- Explain that an inequality is a number sentence that states two quantities are not equal.
Inequalities may be combined by the word ‘‘or’’ or by the word ‘‘and.’’ - Review the information and examples on theworksheet with your students. Make sure they
understand that disjunctions are number sentences joined by ‘‘or’’ and conjunctions are num-
ber sentences joined by ‘‘and.’’- Discuss the examples of the disjunctions and point out that some, such as the third
example, cannot be expressed as one inequality. - Discuss the examples of the conjunctions. Explain that in the first two examples,xis
between two numbers.xis larger than the number on its left (in the combined inequality)
and smaller than the number on its right (in the combined inequality). Note that this is
the preferred way of expressing conjunctions. Also note that if a number is between two
numbers, it excludes the two numbers it is between. - Focus the attention of your students on the last example,− 1 ≥x>−3. This should be
rewritten as− 3 <x ≤−1, the preferred way of writing the expression. Be sure that
your students understand how the inequality was rewritten:− 1 ≥xandx>−3 means
the same as− 3 <xandx≤−1. Therefore, the original inequality can be rewritten as
− 3 <x≤−1.
- Discuss the examples of the disjunctions and point out that some, such as the third
EXTRA HELP:
The combined inequalitya<n<bis a conjunction that meansnis between the numbersaandb.ANSWER KEY:
(1) 3 <y< 7 (2)− 1 ≤x (3)− 4 <x≤ 0 (4) 2 <x< 5 (5)− 1 <y< 1
(6)− 10 <y≤− 3 (7)− 1 ≤x< 4 (8)Impossible
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(Challenge)All real numbers are described because all real numbers are either less than zero or
------------------------------------------------------------------------------------------greaterthanorequaltozero.112 THE ALGEBRA TEACHER’S GUIDE