The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

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WORKSHEET 3.14: SOLVING COMBINED

INEQUALITIES — CONJUNCTIONS

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Two inequalities joined by the word ‘‘and’’ are called a conjunction. To solve a conjunction

follow the steps below:

1. Rewrite each combined inequality as two inequalities joined by the word ‘‘and.’’ The
   entire expression between the inequality symbols is written twice. It is greater than, or
   greater than or equal to, the value on the left, and less than, or less than or equal to,
   the value on the right.


2. Solve each inequality. Remember to change the direction of the inequality symbol if you
   are multiplying or dividing both sides of the inequality by a negative number.


3. Rewrite the solution so that the value of the variable is greater than one numberand
   less than the other.

EXAMPLES

− 1 <x+ 2 < 5 − 4 <− 2 x≤ 10

− 1 <x+2andx+ 2 < 5 − 4 <− 2 xand− 2 x≤ 10

− 3 <xandx< 32 >xandx≥− 5

− 3 <x< 3 − 5 ≤x< 2

DIRECTIONS: Solve each conjunction.

1. − 3 <x− 4 < 7 2. 4 <y− 4 ≤ 5


2. − 6 ≤ 3 +y< 4 4. 2 ≤ 2 y< 10


3. − 1 ≤−x< 3 6.− 3 ≤ 1 −y< 4


4. − 3 ≤ 2 y− 1 < 1 8. 8 <− 2 + 2 x< 16

CHALLENGE:Serena solved 0<− 2 y<4 and found the solution to be

0 <y<−2. This is impossible because a number cannot be less than− 2

and greater than zero. Where did she make her mistake? Find the correct

solution.

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2011 by Judith A. Muschla, Gary Robert Muschla, and Erin Muschla. All rights reserved.