A General Note: Properties of Inequalities
Addition PropertyMultiplication PropertyIf a<b, then a+c<b+c.If a<b and c>0, then ac<bc.If a<b and c<0, then ac>bc.
These properties also apply to
a≤b,
a>b, and
a≥b.
Example 3: Demonstrating the Addition Property
Illustrate the addition property for inequalities by solving each of the following:
a. x−15<4
b. 6≥x−1
c. x+7>9
Solution
The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
a.
x−15<4x−15+15<4+15x<19Add 15 to both sides.
b.
6≥x−16+1≥x−1+17≥xAdd 1 to both sides.
c.
x+7>9x+7−7>9−7x>2Subtract 7 from both sides.
Example 4: Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:
- 3x<6
- −2x−1≥5
- 5−x>10
Solution
Solving this inequality is similar to solving an equation up until the last step.
13−7x≥10x−413−17x≥−4−17x≥−17x≤1Move variable terms to one side of the inequality.Isolate the variable term.Dividing both sides by −17 reverses the inequality.
The solution set is given by the interval
(−∞,1], or all real numbers less than and including 1.
Example 6: Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation:
−43x≥−85+32x.
Solution
We begin solving in the same way we do when solving an equation.
−43x≥−85+32x−43x−32x≥−85−129x−128x≥−85−1217x≥−85x≤−85(−1712)x≤3415Put variable terms on one side.Write fractions with common denominator.Multiplying by a negative number reverses the inequality.
The solution set is the interval
(−∞,3415].