document.write( "Question 1199691: Solve 2x-1 < (x+7)/(x+1) algebraically \n" ); document.write( "

Algebra.Com's Answer #833665 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The partial solution method shown by the other tutor will lead to the wrong answer, because it uses an invalid step.

\n" ); document.write( " $\"2x-1%3C%28x%2B7%29%2F%28x%2B1%29\"$
\n" ); document.write( " $\"2x-1-%28x%2B7%29%2F%28x%2B1%29%3C0\"$

\n" ); document.write( "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.

\n" ); document.write( " $\"%28%282x-1%29%28x%2B1%29-%28x%2B7%29%29%2F%28x%2B1%29%3C0\"$
\n" ); document.write( " $\"%282x%5E2-x%2B2x-1-x-7%29%2F%28x%2B1%29%3C0\"$
\n" ); document.write( " $\"%282x%5E2-8%29%2F%28x%2B1%29%3C0\"$
\n" ); document.write( " $\"%282%28x%5E2-4%29%29%2F%28x%2B1%29%3C0\"$
\n" ); document.write( " $\"%282%28x%2B2%29%28x-2%29%29%2F%28x%2B1%29%3C0\"$
\n" ); document.write( " $\"%28%28x%2B2%29%28x-2%29%29%2F%28x%2B1%29%3C0\"$

\n" ); document.write( "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.

\n" ); document.write( "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).

\n" ); document.write( "ANSWER: (-infinity,-2) U (-1,2)

\n" ); document.write( "Here is a graph showing that $\"2x-1-%28x%2B7%29%2F%28x%2B1%29\"$ is negative on exactly those intervals.

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