Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Solving Inequalities 10-2


Property of Comparison
For all real numbers a and b, one and only one of the following statements is true:

a < b,            a = b,       or a >; b

Transitive Property of Order
For all real  numbers a, b, and c :

1. If a < b and b < c , then  a <; c
2. If a > b, and b > c, then a > c

Addition Property of Order
For all real  numbers a, b, and c :

1. If  a < b, then a + c < b + c
2. If a > b, then a + c > b + c

Multiplication Property of Order


For all real  numbers a, b, and c such that
c > 0 ( c is positive)

1. If a < b, then ac < bc
2. If a > b then ac > bc

c < 0 ( c is negative)

1. If a < b, then ac > bc
2. If a > b, then ac < bc

Multiplying each side of an inequality by a negative number reverses the direction or order of the inequality.

These properties guarantee that the following transformations of a given inequality will always produce an equivalent inequality

1. Substituting for either side of the inequality an expression equivalent to that side
2. Adding to (or subtracting from) each side of the inequality by the same real number
3. Multiplying (or dividing) each side of the inequality by the same positive number.
4. Multiplying  or dividing) each side of the inequality by the same negative number AND  reversing the direction of the inequality. 

To solve an inequality you use the same steps used to solve equations

1. Simplify each side of the inequality as needed
2. Use the inverse operations to undo any additions or subtractions
3. Use the inverse operations to undo any multiplications or divisions

Remember when graphing these inequalities a closed dot indicates that the number is included while an open dot indicates a less than or a greater than.

When working with inequalities it is important to read both the symbols and the words slowly and carefully!!

Word problems involving inequalities also require very careful reading. An important step to solving this type of problem is determining which inequality symbol to use. Before you write your inequality you should be certain that you will use the correct symbol.