solve the inequality

Question

Solve the inequality.  Check your solutions.

 

7−2/b<5/b

 

 

 

Answer 1

You are learning the rules for solving an inequality:

   (1) you may add or subtract the same amount from both sides without changing the inequality, but

   (2) if you multiply or divide by a positive value, it maintains the sense of the inequality, and

   (3) if you multiply or divide by a negative value, it reverses the sense of the inequality.

 

Given:  7−2/b<5/b

           7 < 7/b               (add 2/b to both sides)

           1 < 1/b               (divide both sides by +7)

 

Now, be careful -- we don't know whether b is positive to negative.

 

            b < 1      if b is positive         (multiply both sides by b)

            b > 1      if b is negative        (multiply both sides by b)

 

(p.s., did you notice the word "solutions" in the problem statement?)

 

So, let's pick some values and check.

    For b=2 (a convenient choice), is  7-2/2 < 5/2  ?

                                                     6 <5/2  ?    No.    Good!  2 is outside our solution range.

    For b=-2 (another convenient choice), is  7-2/(-2) < 5/(-2)   ?

                                                                 8 < -5/2    ?   another No.  Good!

 

    Well, we can't test 0 because we would have to divide by 0.

    For b=1,  is 7 -2/1 < 5/1   ?

                       5 < 5  ?     No, but then b=1 also is outside our solution range (we don't allow equal 1 or -1)

    O.K., let's test 1/2.  is 7 - 2/(1/2) < 5/(1/2)  ?

                                     7 - 4 <  10   ?     Yes!

    And, lastly, test -1/2:   Is 7 - 2/(-1/2) > 5/(-1/2)   ?        (careful, b is negative, so use ">")

                                          7 + 4  > 10  ?     Yes!

 

p.s., This exercise will come in very handy when you work with absolute value inequalities.

Answer 2

7 - ( 2 / b) < 5 / b

 

 

We need to find the LCD of the inequality.  The LCD is b.

 

(7b - 2) / b < 5 / b

 

 

The denominators are now the same.  We only need to compare the numerators.

 

7b - 2 < 5

 

 

Add 2 on both sides of the inequality.

 

7b < 7

 

 

Divide both sides of the inequality by 7.

 

b < 1

 

 

The solution is all values of b less than 1.