Teaching Notes 3.12: Solving Inequalities with Variables
on One Side
The most common error students make in solving inequalities with a variable on one side is to
forget to change the direction of the inequality symbol when they multiply or divide both sides
of the equation by a negative number. When students understand why they need to change the
direction of the inequality symbol, they are less likely to make this mistake.- Discuss the ways students can write equivalent inequalities:
- Add the same number to or subtract the same number from both sides of the inequality.
Example: 3 <4 can be rewritten as 3+ 5 < 4 +5or3− 1 < 4 −1. - Multiply or divide both sides of the inequality by the same positive number.Example:
3 <4 can be rewritten as 3× 5 < 4 ×5or15<20.
In each case, the inequalities are equivalent.
- Add the same number to or subtract the same number from both sides of the inequality.
- Ask your students to consider what happens if you multiply both sides of an inequality by
a negative number. Offer this example: 3<4but3×(−2)> 4 ×(−2). Emphasize that the
direction of the inequality symbol changed. Instructyour students to sketch a number line
to recognize this relationship. Explain that the direction of an inequality symbol would also
change if they were to divide both sides of an inequality by a negative number. Provide more
examples if necessary. - Reinforce the multiplication axiom of order:
- Ifa<b,thenac
b,thenac>bc,wherec>0. - Ifa<b,thenac>bcand ifa>b,thenac<bc,wherec<0.
- Ifa<b,thenac
- Review the information and the examples on the worksheet with your students.
EXTRA HELP:
Multiplying or dividing both sides of an inequality by zero will never produce an equivalent
inequality.ANSWER KEY:
(1)x> 7 (2)y≤ 2 (3)x≤− 5 (4)x>− 8 (5)y> 9 (6)x≤ 4 (7)y> 10 (8)x< 100
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(Challenge)Yes, he is correct. The variable may be placed on the left side of the inequality.− 2 >x
isthesameasx<−2.
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