
Solving Inequalities
Properties of Inequalities For all real numbers a,
b, c:
Rule 1 
If a < b, then b > a 
Rule 2 
If a < b, then a > b 
Rule 3 
If a < b, then for all real numbers c, a + c < b
+ c 
Rule 4 
If a < b, then for all POSITIVE c, ac < bc 
Rule 5 
If a < b, then for all NEGATIVE c, ac > bc 
Solution sets to a linear inequality can be shown using
table 2.6(pg 174)
Steps to Solving a Linear Inequality
(The < sign here could be ANY inequality.) 
1. Eliminate fractions by multiplying both sides
by the least common denominator. KEEP THE LCM POSITIVE! 
2. Remove grouping symbols using the distributive
property 
3. Simplify each side by combining like terms.
(ax + b < cx + d) 
4. Use addition property of equality to isolate
the variable. (gx + b) < d 
5. Use addition property of equality to isolate
the constant term. to get the
equation into the form (gx < h). 
6. If g > 0, use the multiplication property of
equality to solve for x (x < k). 
7. If g < 0 use the multiplication property of
equality to solve for x and CHANGE the direction of the inequality. 
8. Check your solution in the original inequality
by substitution. 
