Rational Inequality

Question

4/(c-2)-3/c <0

Answer 1

Cross multiply after addition across:

4/(c-2)-3/c <0

          +3/c    +3/c

4/(c-2)       < 3/c

   *(c-2)         (c-2)

4               < (3c-6)/c

  *c                 *c

4c             < 3c-6

 -3c              -3c

 c              < -6

  c < -6

 

The thing to watch out for in these kinds of problems is that if you ever multiply both sides by a negative, the inequality changes sides.

 

c   <  -6

*-1   *-1

-c  >  6

 

Answer 2

4/(c-2) - 3/c < 0

 

Simplify the left side:  [4c - 3(c-2)]/[c(c-2)] < 0

 

                                (c + 6)/[c(c-2)] < 0

 

Numerator = 0 when c = -6

Denominator = 0 when c = 0 or c = 2

 

Use the values obtained above to divide the number line into a collection of intervals and choose a "test value" from each interval:

 

_________l_________l__________l____________

             -6               0               2

 

Interval          Test value           sign of (c+6)/[c(c-2)] at test value

 

c<-6                    -7                          -

-6<c<0                -1                          +

0<c<2                  1                          -

c>2                      3                          +

 

Solution set:  (-∞,-6) ∪ (0,2)