Question 1209292
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The problem would be far more educational if the answers were not so ugly....<br>
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.<br>
{{{(3-z)/(z+1)-2-1/(z-7)>=0}}}<br>
{{{((3-z)(z-7)-2(z+1)(z-7)-1(z+1))/((z+1)(z-7))>=0}}}<br>
{{{(3z-21-z^2+7z-2z^2+12z+14-z-1)/((z+1)(z-7))>=0}}}<br>
{{{(-3z^2+21z-8)/((z+1)(z-7))>=0}}}<br>
The expression is undefined where the denominator is zero -- at z=-1 and z=7.<br>
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<br>
{{{A=(21-sqrt(345))/6}}} = approximately 0.4043041
{{{B=(21+sqrt(345))/6}}} = approximately 6.5956959<br>
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.<br>
(1) (-infinity,-1) negative
(2) (-1,A] positive or zero
(3) (A,B) negative
(4) [B,7) zero or positive
(5) (7,infinity) negative<br>
ANSWER: The inequality is true on the two intervals (-1,A] and [B,7)<br>