Properties of Inequalities:
less than or equal to £
greater than or equal to ³
Addition Property of Inequality Adding both sides of an inequality with a  positive number  does not change the inequality sign  If a < b, then a + c < b + c
If a £ b, then a + c £ b + c
If a ³ b, then a + c ³ b + c

Subtraction Property of Inequality Subtracting both sides of an inequality with a  positive number  does not change the inequality sign  If a < b, then a  c < b  c If a £ b, then a  c £ b  c
If a ³ b, then a  c ³ b  c

Multiplication Property of Inequality Mulitplying both sides of the inequality with a  positive number  does not change the inequality sign  If a < b AND c is positive, then ac < bc
If a £ b AND c is positive, then ac £ bc

Division Property of Inequality  positive number  does not change the inequality sign  If a < b AND c is positive, then a/c < b/c
If a ³ b AND c is positive, then a/c ³ b/c

Multiplication Property of Inequality  negative number  changes the inequality sign  If a < b AND c is negative, then ac > bc
If a £ b AND c is negative, then ac ³ bc 
Division Property of Inequality  negative number  changes the inequality sign  If a < b AND c is negative, then a/c > b/c
If a £ b AND c is negative, then a/c ³ b/c 
If both sides of an inequality are positive and n is a positive integer  nth power or nth root  does not change the inequality  a > b a^{2} > b^{2}
Öa > Öb

Reciprocal  On both sides of the inequality  changes the inequality sign  1/a > 1/b
a < b

 Addition/Subtraction Property of Inequality:
Adding or subtracting a positive number from both sides of an inequality does not change the inequality sign.
If a < b, then a + c < b + c
If a < b, then a  c < b  c
If a £ b, then a + c £ b + c
If a ³ b, then a + c ³ b + c
If a £ b, then a  c £ b  c
If a ³ b, then a  c ³ b  c
 Multiplication/Division Property of Inequality:
Multiplying and dividing both sides of an inequality by a positive number does not change the inequality sign. This is not true for a negative number b.
If a < b AND c is positive, then ac < bc
If a < b AND c is positive, then a/c < b/c
If a £ b AND c is positive, then ac £ bc
If a ³ b AND c is positive, then a/c ³ b/c
 Multiplying or dividing both sides of an inequality by a negative number changes the inequality sign.
If a < b AND c is negative, then ac > bc
If a < b AND c is negative, then a/c > b/c
If a £ b AND c is negative, then ac ³ bc
If a £ b AND c is negative, then a/c ³ b/c
 If both sides of an inequality are positive and n is a positive integer, then the inequality formed by the nth power or nth root of both sides does not change the inequality.
Example: 9 > 6
9^{2} > 6^{2}
81 > 36 That is still true.
Example: Ö9 > Ö6
3 > 2.45
 Taking the reciprocal on both sides of the inequality changes the inequality sign.
Example:
1/2 > 1/4
Taking reciprocal both sides changes the inequality
2 < 4
Example:
Solve 5x + 3 < 10.
Solution:
Given that 5x + 3 < 10.
Add 3 on both sides, we get
5x + 3  3 < 10  3
5x < 7
Dividing both sides by 5, we get
5x/5 < 7/5
x < 7/5
Therefore, any number x < 7/5 is a solution.
Example:
Solve 3d  2(8d  9) > 2d  4
Solution:
3d  2(8d  9) > 2d  4 Original inequality
3d  16d + 18 > 2d  4 Distributive property
 13d + 18 > 2d  4 Combining like terms
 13d + 18 + 13d > 2d  4 + 13d Adding 13d both sides
18 > 11d  4 Simplify
18 + 4 > 11d  4 + 4 Adding 4 each side
22 > 11d Simplify
22/11 > 11d/11 Divide each side by 11
2 > d Simplify
That is d > 2
Therefore the solution set is {dd < 2}
Directions: Solve the following inequations. Also write at least 10 examples of your own.
