Solving Linear Inequalities
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Solving linear inequalities is the same as solving linear equations...

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with one very important exception...
 

when you multiply or divide an inequality by a negative value, it changes the direction of the inequality. 


Inequalities with one variable:

Consider:

Look at this true statement:
Suppose we multiply both sides by  -1.

What is the relationship between these two numbers ?

          5 > 3
(-1)(5) ? (3)(-1)
    -5    ?   -3   


ANS:  -5 is less than -3 because it is further to the left on the number line.

     
     -5 < -3
So, we must change the direction of the inequality when we multiply (or divide) by a negative number in order to get the correct answer.
 

Before we begin our example problems, refresh your memory on what each inequality symbol means.
It is helpful to remember that the "open" part of the inequality symbol (the larger part) always faces the larger quantity.

SYMBOL

MEANING

less than

greater than

less than or equal to

greater than or equal to

 

Example 1

 Solve and graph the solution set of:   2x - 6 < 2

Add 6 to both sides.
Divide both sides by 2.

Open circle at 4 (since x can not equal 4) and an arrow to the left (because we want values less than 4).

2x - 6 < 2
     2x < 8
       x < 4 

 

Example 2

 Solve and graph the solution set of:   5 - 3x 13 + x

Subtract 5 from both sides.
Subtract x from both sides.

 Divide both sides by -4, and don't forget to change the direction of the inequality !
(We divided by a negative.)
 

5 - 3x 13 + x
    -3x 8 + x
    -4x 8
        x -2
Closed circle at -2 (since x can equal  -2) and an arrow to the right (because we want values larger than -2).

 

 

Example 3

Solve and graph the solution set of:  3(2x + 4) > 4x + 10 

Multiply out the parentheses.
Subtract 4x from both sides.
Subtract 12 from both sides.

 Divide both sides by 2, but don't change the direction of the inequality, since we didn't divide by a negative.
 		 3(2x + 4) > 4x + 10
  6x + 12 > 4x + 10
  2x + 12 > 10
       2x > -2
         x > -1 
Open circle at -1 (since x can not equal -1) and an arrow to the right (because we want values larger than -1).


See how to use your
TI-83+/84+ graphing calculator  with linear inequalities.
Click calculator.