154 CHAPTER 2 Linear Equations and Inequalities in One Variable
OBJECTIVE 3 Use the multiplication property of inequality. Consider the
true inequality Multiply each side by the positive number 2.
Multiply each side by 2.
TrueNow multiply each side of by the negative number
Multiply each side by 5.
FalseTo get a true statement when multiplying each side by 5, we must reverse the
direction of the inequality symbol.
Multiply by 5. Reverse the symbol.- 15 7- 35 True
- 5132 7- 5172 -
3 67
-
- 15 6- 35
- 51326 - 5172 -
3 67
367 - 5.
6 614
213262172
3 67
36 7.
Multiplication Property of InequalityIf A, B, and Crepresent real numbers, with and
1. if Cis positive,then the inequalities
and
have exactly the same solutions;
2. if Cis negative,then the inequalities
and
have exactly the same solutions.
That is, each side of an inequality may be multiplied by the same positive
number without changing the solutions. If the multiplier is negative, we must
reverse the direction of the inequality symbol.
ABC
A<B AC<BC
CZ0,
As with the multiplication property of equality, the same nonzero number may be
divided into each side of an inequality.
Note the following differences for positive and negative numbers.
1.When each side of an inequality is multiplied or divided by a positive number,the
direction of the inequality symbol does not change.
2.Reverse the direction of the inequality symbol ONLY when multiplying or
dividing each side of an inequality by a NEGATIVE NUMBER.
NOTE The above illustrations began with the inequality a true statement in-
volving two positive numbers. Similar results occur when one or both of the numbers
is negative. Verify this with
by multiplying each inequality first by 2 and then by
These observations suggest the multiplication property of inequality.
- 5.
- 36 7, 3 7-7, and - 7 6- 3
36 7,
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