Question 1199691
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The partial solution method shown by the other tutor will lead to the wrong answer, because it uses an invalid step.<br>
{{{2x-1<(x+7)/(x+1)}}}
{{{2x-1-(x+7)/(x+1)<0}}}<br>
You can't multiply everything by (x+1) here, because for some values of x (x+1) is negative, and for one particular value of  (x+1) is zero.  We need to keep the denominator in our solution.<br>
{{{((2x-1)(x+1)-(x+7))/(x+1)<0}}}
{{{(2x^2-x+2x-1-x-7)/(x+1)<0}}}
{{{(2x^2-8)/(x+1)<0}}}
{{{(2(x^2-4))/(x+1)<0}}}
{{{(2(x+2)(x-2))/(x+1)<0}}}
{{{((x+2)(x-2))/(x+1)<0}}}<br>
The value of the expression on the left changes sign only when one of the factors in the numerator or denominator changes sign; that happens only at x = -2, x = -1, and x=2.<br>
Looking at the intervals determined by those three values of x, you will see that the expression is negative on (-infinity,-2) and (-1,2); it is positive or zero on [-2,-1) and on [(2,infinity).<br>
ANSWER: (-infinity,-2) U (-1,2)<br>
Here is a graph showing that {{{2x-1-(x+7)/(x+1)}}} is negative on exactly those intervals.<br>
{{{graph(400,400,-5,5,-50,50,((2x-1)(x+1)-(x+7))/(x+1))}}}<br>