Inequalities Section 10.2 Solving Inequalities

Property of Comparison For all real numbers a and b, one and only one of the following must be true: a<b a=b or a>b

Transitive Property For all real numbers a, b and c: If a<b and b<c, then a<c If c>b and b>a, then c>a

Addition Property of Order For all real numbers a, b and c If a<b, then a+c<b+c If a>b, then a+c>b+c We can add the same quantity to both sides of the equal sign and we don’t change the order relationship

Multiplication Property of Order For all real numbers a, b and c such that c>0 (c is positive) If a<b, then ac<bc If a>b, then ac>bc We can multiply both sides by the same positive quantity and we don’t change the order relationship

Multiplication Property of Order For all real numbers a, b and c such that C<0 (c is negative) If a bc If a>b, then ac<bc We can multiply both sides by the same negative quantity but we must reverse the order relationship (flip the inequality sign)

Transformations that Produce an Equivalent Inequality Substituting an equivalent expression on either side of the inequality Adding or subtracting the same quantity on both sides of the inequality Multiplying or dividing both sides of the inequality by the same POSITIVE quantity Multiplying or dividing both sides of the inequality by the same NEGATIVE quantity, PROVIDED YOU REVERSE THE DIRECTION OF THE INEQUALITY (flip the inequality sign)

Example 1

Example 2

This is false since zero is really less than eight, so the original inequality has no solution. Example 3

This is true since five is less than six, so the original inequality is true for all real numbers. Example 4

Assignment see syllabus