Solving Inequalities
(including graphing and interval notation)

Inequalities are solved in the same manner as equations.  The only exception in solving inequalities is that if you multiply or divide the sides of an inequality by a negative number you must reverse the direction of the inequality.
 

Solve:               

                                   …Distribute

                                              …the results

                       …Collect

                       …Addition (and subtraction)

                                                                  …the results

                             …Division

                                …the solution, notice that we divided by a negative; therefore, we must reverse the direction of the inequality

In the next example, do not multiply (or divide) by a negative and therefore do not reverse the direction of the inequality.

Solve:                                              

               …Multiply the terms by the reciprocal                      

                         …since we multiplied by a positive number, we do not reverse the inequality symbol

Switching the direction of the inequality is based solely on the last step and whether or not you multiply or divide by a negative number.  (i.e. Is the coefficient to the variable negative or not?  If it is negative, reverse the inequality in the last step … If it is not negative, leave the inequality in its original position.)

To graph an inequality, it is helpful to solve for the variable on the left.  The graphing makes much more sense that way.  For example, if we solved an inequality and the answer was , what would that mean?  8 is less than or equal to x?  This really makes no sense!  However, turn that around and , which is totally logical …  x is greater than or equal to 8.  Notice that the smaller point of the inequality is pointing to the 8 in each of the above inequalities and therefore is the same answer.

It is much easier to express the answer in a graph and in interval notation when the variable is on the left.   Use a bracket when the inequality is  and a parenthesis when the inequality is < or >.  When using the  or  symbols (infinity or negative infinity) always use a parenthesis.

Graph the solution  and write in interval notation:           

(graph)                                     [          

                                                8

(interval notation)                      

 

Graph the solution x < 5 and write in interval notation:

 (graph)                                           )             

                                                        5

(interval notation)                     

Notice that the arrow to the answer would point in the direction of the inequality, assuming that the inequality is solved so that the variable is on the left (as discussed above.)

Solving compound inequalities.  To solve a compound inequality (an inequality with 3 parts), solve it in the same manner as if solving an equation, but remember to always keep the variable in the middle.  Whatever operations are performed on the middle to isolate the variable, should be performed identically on the two outside parts. 

Solve:                                    

                                                         … add 1 to all three sections

                                                              … divide all three terms by 2

                                                                   … the answer                                   

(graph)                               [               )                                                       

                                         -3                 8

(interval notation)                               


Notice, once again, the use of the bracket and parenthesis in graphing and in interval notation.  The solution to this compound inequality lies between the values of the two numbers.  Also notice that this is a continuous inequality; in other words, the flow of the numbers is from smallest to largest (from left to right) as it would be on a number line.  Look once more at a simple compound inequality and the changes that have to be made at the end to insure that the flow of the numbers from left to right is from smallest to largest.

Solve:                       

                                                                 … distribute

                                                                 … subtract 5 from all three sections

                                                                     … divide all terms by

                                                                      … notice the direction of the inequalities     
                                                                                      has been switched, since we divided by a
                                                                                      negative number.  The flow of  numbers
                                                                                      from left to right makes no sense now.

                                                                   Turn everything around so that the 
                                                                                      smallest number possible is on the left and
                                                                                      the largest number possible is on the right.
                                                                                      Now the inequality is continuous.

(graph)                               (             ]                                                         

                                         -3              14

(interval notation)                   

 

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 09-Jan-2014