SOLUTION: Solve the inequality (3 - z)/(z + 1) \ge 2 + 1/(z - 7). Write your answer in interval notation.

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Question 1209292: Solve the inequality
(3 - z)/(z + 1) \ge 2 + 1/(z - 7).
Write your answer in interval notation.
Found 3 solutions by mccravyedwin, Edwin McCravy, greenestamps:
Answer by mccravyedwin(408)   (Show Source): You can put this solution on YOUR website!


Draw the graph of the left side and the right side:



The exact solutions are ugly.

I found the approximate solutions of their points of
intersection with a TI-84. They are 

0.4043040632... and 6.595695937...

There are asymptotes at -1 and 7.
The red curve is the graph of the left side.
The green curve is the graph of the right side.
The blue vertical lines are the asymptotes.

The red curve is above or intersecting the green curve
between the asymptote at -1 and the smaller solution.
So that part is the interval:

(-1, 0.4043040632...]

The red curve is above or intersecting the green curve
again between the larger solution and the asymptote at 7
So that part is the interval:
   
[6.595695937..., 7)

So the solution is 

(-1, 0.4043040632...] U [6.595695937..., 7)

Edwin
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


The problem would be far more educational if the answers were not so ugly....

To solve an inequality like this, get everything on the left with 0 on the right and write the expression on the left as a single rational expression.









The expression is undefined where the denominator is zero -- at z=-1 and z=7.

The expression value is zero where the numerator is zero. The quadratic in the numerator has irrational roots, which for simplicity I will represent with A and B. The values of those roots are

= approximately 0.4043041
= approximately 6.5956959

The four zeros of the numerator and denominator break the number line into 5 intervals. Use test points or other methods to determine the intervals on which the expression value is greater than or equal to 0.

(1) (-infinity,-1) negative
(2) (-1,A] positive or zero
(3) (A,B) negative
(4) [B,7) zero or positive
(5) (7,infinity) negative

ANSWER: The inequality is true on the two intervals (-1,A] and [B,7)
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