
A quick review of working with linear inequalities:

a > b ; a is strictly greater than b 
a < b ; a is striclty less than b 
Note: a > b and a < b are called "strict inequalities". 
a b ; a is greater than or equal to b 
a b ; a is less than or equal to b 
a ≠ b ; a is not equal to b 
Hint: The "open" (larger) part of the inequality symbol always faces the larger quantity. 

Inequality Important Rule: 
The process of solving a linear inequality is the same as solving a linear equation, except ...
... when you multiply (or divide) an inequality by a negative value,
you must change the direction of the inequality.

Check out the "Refresh Section" if you need more review on working with linear inequalities. 
Let's start by justifying why the "Inequality Important Rule" (stated above) is true.
Statement: If you multiply (or divide) both sides of an inequality by a negative value,
you will need to change the direction of the inequality.
1. If a > b, then a < b. 
1. Translation of statement (mult. by 1). 
2. a > b 
2. Start with the "given" a > b. 
3. a  b > b  b 
3. Subtract b from both sides. 
4. a  b > 0 
4. Additive Inverse Property b  b = 0 
5. a  a  b > 0  a 
5. Subtract a from both sides. 
6. 0  b > a 
6. Additive Inverse, Additive Identity 
7. b > a 
7. Additive Identity Property 
8. a < b 
8. Read in reverse (converse) 
Basic Properties of Inequalities: (properties apply to )
If a > b, then a + c > b + c. 
Addition Property of Inequality 
If a > b, then a  c > b  c. 
Subtraction Property of Inequality 
If a > b and c > 0, then ac > bc. 
Multiplication Property of Inequality

Notice that c > 0.
If c < 0, reverse the direction of the final inequality. 
If a > b and c > 0, then . 
Division Property of Inequality 
Solving Linear Inequalities:
For more basic problems on solving inequalities, see Refresher section.


For calculator help with linear inequalities
click here. 


Solve and graph the solution set of: 
Proceed as you would when solving a linear equation:
Add 6 to both sides.
Multiply both sides by .
Note: The direction of the inequality stays the same since we did NOT multiply by a negative value.
Graph using a closed circle for 6 (since x can equal 6) and an arrow to the left (since our symbol is less than or equal to).

Other notations:

Solve and graph the solution set of: 5(x  3) > 10 
Distribute (remove the parentheses).
Add 15 to both sides.
Divide both sides by 5.
Note: The direction of the inequality stays the same since we did NOT multiply by a negative value.
Graph using a open circle for 5 (since x can NOT equal 5) and an arrow to the right (since our symbol is greater than). 
Other notations:

Solve and graph the solution set of: 
Multiply both sides by the least common denominator, which is 8.
Divide both sides by 1, and flip the direction of the inequality.
Note: The direction of the inequality was reversed since we divided by a negative value (1).
Graph using a closed circle for 2 (since x can equal 2) and an arrow to the left (since our symbol is less than or equal to). 
Other notations:

Solve and graph the solution set of: 4(x  1) > 3(x  2) 
Distribute across both sets of parentheses.
Subtract 3x from both sides.
The solution is easier if you move the smaller x value.
Add 4 to both sides.
Note: The direction of the inequality stays the same since we did NOT multiply or divide by a negative value.
Graph using an open circle for 2 (since x can not equal 2) and an arrow to the right (since our symbol is greater than). 
Other notations:

Solve and graph the solution set of: 4(2x + 1) < 8x + 4 
Distribute across the parentheses.
Subtract 8x from both sides.
Notice the FALSE result.
The solution to this inequality is the "empty set" (). There are no xvalues which will make this inequality true.
There is nothing to be graphed. 
This tells us that NO xvalues will make this inequality true.

In this problem, the left side of the inequality is simply another way of writing the right side of the inequality. The two sides are EQUAL to one another. Since a quantity can never be less than itself, this inequality is never true. 
Solve and graph the solution set of: 
Proceed as you would when solving a linear equation:
Subtract 3 from both sides.
Multiply both sides by 2.
The solution to this inequality is all real numbers except 2. The number 2 is the only value which makes this inequality false, since:

Other notations:

Solve and graph the solution set of: 
Hint: In this problem, the negative signs appear in front of the two fractional terms. You can place the negative sign in the numerator, or the denomoniator, since
This example places the negative sign in the numerators.
Multiply all terms by the least common denominator (which is 6).
Add 4x to both sides.
The solution is easier if you move the smaller x value.
Be sure to write the 0 on the right side.
Subtract 12 from both sides.
Divide both sides by 3.

Other notations:


Solve for x: ax > 6a 
At first glance, it seems that dividing both sides by a will solve the problem.
BUT ...
We just don't know if a is a positive value or a negative value. If a is positive, our solution of x > 6 will be true.
But if a is negative, we need to reverse the direction of the inequality, and the solution is x < 6.
Since we do not know the value of a, we cannot state a solution to this problem.
Therefore, there is NO SOLUTION!

_________________________________

An inequality cannot be solved by dividing by a "variable" (a letter) if you do not know that the "variable" is either always positive or always negative.
(And, of course, division by zero is never allowed.) 
NOTE: The reposting of materials (in part or whole) from this site to the Internet
is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use". 

