Question 253049
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Look for counterexamples for each answer choice -- that is, positive values of x for which the expression is an integer.<br>
(A) {{{ sqrt(x-1) }}}
x=1 yields value 0, which is an integer<br>
(B) {{{sqrt(x^2 - 1)}}}
again x=1 yields value 0<br>
(C) {{{ (x+4) / (x-2) }}}
x=3 makes the denominator 1 and so yields an integer value<br>
(E) {{{ 5/(x+1) }}}
x=4 yields value 1, which is an integer<br>
(D) {{{ (x+4) / (x+2) }}}<br>
I left this for last, because (D) is the answer.<br>
To see this, note that {{{ (x+4) / (x+2) = 1+2/(x+2)}}}<br>
In that reformatted expression, 1 is an integer, so {{{2/(x+2)}}} must be an integer.  But {{{2/(x+2)}}} is an integer only when (x+2) is a factor of 2 -- i.e., when (x+2) is -2, -1, 1, or 2 -- and x is not a positive integer for any of those.<br>
ANSWER: (D)<br>