Question 1183772
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(Revised/expanded response, following comment from student....)<br>
The solution from the other tutor arrives at the correct solution.  However, the method of solution is deficient; and the limited explanation of HOW to solve the problem makes the response of little use to the student.<br>
The function changes behavior at x=-7 and x=4; so three intervals must be examined for solutions: x less than -7, x between -7 and 4, and x greater than 4.<br>
(1) If x is less than -7, then
x+7 is negative, so {{{abs(x+7)=-(x+7)=-x-7}}}
and x-4 is also negative, so {{{abs(x-4)=-(x-4)=-x+4}}}<br>
and then the equation is
{{{(-x-7)+(-x+4)=13}}}
{{{-2x-3=13}}}
{{{-16=2x}}}
{{{x=-8}}}<br>
(2) If x is between -7 and 4, then
x+7 is positive, so {{{abs(x+7)=x+7}}}
and x-4 is negative, so {{{abs(x-4)=-(x-4)=-x+4}}}<br>
and then the equation is
{{{(x+7)+(-x+4)=13}}}
{{{11=13}}}<br>
That equation is obviously never true, so there is no solution in the interval between -7 and 4.<br>
(3) If x is greater than 4, then
x+7 is positive, so {{{abs(x+7)=x+7}}}
and x-4 is also positive, so {{{abs(x-4)=x-4}}}<br>
and then the equation is
{{{(x+7)+(x-4)=13}}}
{{{2x+3=13}}}
{{{2x=10}}}
{{{x=5}}}<br>
So there are two solutions: x=-8 and x=5.<br>
Here is a graph of the two functions {{{abs(x+7)+abs(x-4)}}} and {{{13}}}; the two graphs intersect at x=-8 and x=5.<br>
{{{graph(400,200,-10,10,-5,20,abs(x+7)+abs(x-4),13)}}}<br>