Teaching Notes 3.15: Solving Combined
Inequalities—Disjunctions
The first step to solving a disjunction is to write two separate inequalities. The solution of the
disjunction is a solution to at least one of the inequalities (meaning one or the other or both are
true). Students sometimes confuse this process with that ofsolving a conjunction, whose solution
is a solution of both of the inequalities.- Explain that a disjunction is a special type of combined inequality in which two inequalities
are combined by the word ‘‘or.’’ Note that this is different from a conjunction in which two
inequalities are combined by the word ‘‘and.’’ - Note that disjunctions can be written as two inequalities. Each inequality must be solved
independently. - Review the information and example on the worksheet with your students. Emphasize that
students must solve both inequalities. Make sure that they understand the steps for solving
inequalities. Encourage your students to sketch a number line to help them recognize the
relationship. - Remind your students that they must change the direction of the inequality symbol if they
divide both sides of an inequality by a negative number. Depending on your students’ abili-
ties, you may find it helpful to review 3.12: ‘‘Solving Inequalities with Variables on One Side.’’
Encourage your students to check their solutions.
EXTRA HELP:
The solution of a disjunction will never be expressed with the word ‘‘and.’’ANSWER KEY:
(1)y≤−3ory≥ 1 (2)x<2orx> 14 (3)y≤−9ory≥− 1
(4)x<4orx> 8 (5)x<−5orx> 4 (6)x≤1orx> 5
(7)x>9orx<− 3 (8)x<5orx> 7 (9)x<−3orx> 5
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(Challenge)Eva is correct.x=4 is another way of writing the solutionx>4orx<4.
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