Solve the given inequality

AI Thread Summary
The discussion focuses on solving the inequality (x+1)(x+4)/((x-1)(x-2)) - 2 < 0 using algebraic methods. Key points include identifying vertical asymptotes at x=1 and x=2, and determining critical values from the numerator, leading to the inequality -x^2 + 11x < 0. The analysis reveals that the function changes sign at x=0, 1, 2, and 11, with the solution set being x < 0, 1 < x < 2, and x > 11. The conversation clarifies that sign changes can be assessed without taking the derivative, emphasizing the behavior of the function around critical points.
chwala
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Homework Statement
see attached
Relevant Equations
inequalities
I am interested in an algebraic approach.

1712658207073.png


My lines are as follows;

##\dfrac{(x+1)(x+4)}{(x-1)(x-2)} -2<0##

##\dfrac{(x^2+5x+4) - 2(x-1)(x-2)}{(x-1)(x-2)} <0##

The denominator will give us the vertical asymptotes ##x=1## and ##x=2##

The numerator gives us,

##x^2+5x+4-2x^2+6x-4 <0##

##-x^2+11x<0##

##x(-x+11)<0##

##x=0## is critical value ##⇒x<0## and the other critical value is ##x=11## ##⇒x>11##. On checking together with the asymptotes, we end up with
##x<0, 1<x<2 ## and ##x>11##.
Any insight or positive criticism is welcome.
 
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You said "the numerator gives us" but the things that follows is only true if the denominator is positive, which you haven't explicitly handled i think?
 
You have converted the original inequality into
$$\dfrac{x(-x+11)}{(x-1)(x-2)} <0$$
This means that either
$$x(-x+11)<0, (x-1)(x-2)>0$$ or $$x(-x+11)>0, (x-1)(x-2)<0$$
These two options further split:
$$x<0, (-x+11)>0, (x-1)>0,(x-2)>0$$ $$x>0, (-x+11)<0, (x-1)>0,(x-2)>0$$ $$x<0, (-x+11)>0, (x-1)<0,(x-2)<0$$ $$x>0, (-x+11)<0, (x-1)<0,(x-2)<0$$ $$x<0, (-x+11)<0, (x-1)<0,(x-2)>0$$ $$x>0, (-x+11)>0, (x-1)<0,(x-2)>0$$ $$x<0, (-x+11)<0, (x-1)<0,(x-2)>0$$ $$x>0, (-x+11)>0, (x-1)<0,(x-2)>0$$
The first inequality gives $$x<0,x<11,x>1,x>2$$ which is empty.
Etc.
 
We require <br /> f(x) = \frac{x(11-x)}{(x-1)(x - 2)} &lt; 0. The numerator and denominator have no common linear factors, so f changes sign whenever a linear factor of either the numerator or the denominator changes sign. Thus we have sign changes at x = 0, 1, 2, 11.

Starting at x = -\infty, we see that f(x) \sim -1, so f(x) &lt; 0 for x &lt; 0 and 1 &lt; x &lt; 2 and x &gt; 11.
 
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pasmith said:
We require <br /> f(x) = \frac{x(11-x)}{(x-1)(x - 2)} &lt; 0. The numerator and denominator have no common linear factors, so f changes sign whenever a linear factor of either the numerator or the denominator changes sign. Thus we have sign changes at x = 0, 1, 2, 11.

Starting at x = -\infty, we see that f(x) \sim -1, so f(x) &lt; 0 for x &lt; 0 and 1 &lt; x &lt; 2 and x &gt; 11.
For one to use sign change one has to take the derivative of ##f(x)## with respect to ##x##?
 
chwala said:
For one to use sign change one has to take the derivative of ##f(x)## with respect to ##x##?
No, you don't need to take the derivative. For example, looking at the x - 1 factor in the denominator, it will change sign for x on either side of 1, but close to 1. The factors in the numerator, x and 11 - x don't change sign. The other factor in the denominator, x - 2, also doesn't change sign.

Since no other factors change sign, the y value changes sign for values near 1, but on either side of 1. This can be seen by looking at the graph you attached in the OP.
 
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