Question 1206574
<br>
{{{abs(sqrt(x+3)-(7+x)/4)<0.5}}}<br>
{{{-0.5<sqrt(x+3)-(7+x)/4<0.5}}}<br>
{{{-0.5<sqrt(x+3)-(7+x)/4}}} and {{{sqrt(x+3)-(7+x)/4<0.5}}}<br>
It turns out that the maximum value of {{{sqrt(x+3)-(7+x)/4}}} is 0, so we only need to solve the inequality {{{-0.5<sqrt(x+3)-(7+x)/4<0}}}.  If you try to solve the inequality {{{sqrt(x+3)-(7+x)/4<0.5}}} you will find that there are no real solutions.<br>
Solve the corresponding EQUATION to find the endpoints of the solution interval.<br>
{{{-0.5=sqrt(x+3)-(7+x)/4}}}<br>
Multiply everything by 4 to clear fractions
{{{-2=4sqrt(x+3)-7-x}}}<br>
Isolate the term with the radical
{{{x+5=4sqrt(x+3)}}}<br>
Square both sides to eliminate the radical
{{{x^2+10x+25=16x+48}}}
{{{x^2-6x-23=0}}}<br>
The quadratic does not factor; use the quadratic formula
{{{x=(6+sqrt(36+92))/2}}} or {{{x=(6-sqrt(36+92))/2}}}
{{{x=(6+8sqrt(2))/2}}} or {{{x=(6-8sqrt(2))/2}}}
{{{x=3+4sqrt(2)}}} or {{{x=3-4sqrt(2)}}}<br>
To 3 decimal places, the solutions are -2.657 and 8.657.<br>
A graph....<br>
{{{graph(400,400,-10,10,-1,1,-0.5,0.5,abs(sqrt(x+3)-(7+x)/4))}}}<br>