Question : If $16-x^2>|x-a|$ is to be satisfied by at least one negative value of x, then the complete set of values of 'a' is ?
Attempt :
We can write the inequality as :
$$x^2-16 < x-a < 16-x^2$$
$$\Rightarrow x^2-x+(a-16) < 0 \ldots (i)$$and$$ x^2+x-(a+16) < 0 \ldots (ii)$$
For the quadratic inequality (ii), the vertex of the parabola occurs at $x=-1/2$
So it suffices that discriminant $D>0$ for the inequality to be true for atleast one negative value of $x$
However for inequality (i) I'm not able to figure out the conditions required for it to be true. Any help is appreciated. Thanks
Best Answer
https://www.desmos.com/calculator/eupzet7eav
The Graph of $y=|x-a|$ is V-shaped. If $a$ is positive, $a$ must take the value such that the $y$-intercept of $y=|x-a|$ is less than the $y$-intercept of $y=16-x^2$, i.e. $a<16$.
If $a$ is negative, $a$ must take the value such that $y=x-a$ cut $y=16-x^2$ at two distinct points, i.e. $a>-16.25$
Therefore, $-16.25<a<16$.