Question 1183959
<br>
By far the easiest way to find the solution set is with a graphing calculator.  Graph the two functions<br>
{{{y=abs(1-4x^2)-abs(11x-12-2x^2)}}} and {{{y=1}}}<br>
and see where the curve is below y=1.<br>
{{{graph(400,400,-10,2,-30,10,abs(1-4x^2)-abs(11x-12-2x^2),1)}}}<br>
Visually, the solution is the interval between about -6.6 and +1.  A graphing calculator gives the approximate interval to several decimal places as (-6.566084, 1.0660844).<br>
An exact algebraic solution requires a lot of work....<br>
The behavior of the function on the left changes when either of the expressions in absolute value changes sign.<br>
{{{1-4x^2 = (1-2x)(1+2x)}}}<br>
The behavior changes are x=-1/2 and x=1/2.<br>
{{{abs(1-4x^2)=1-4x^2}}} on the interval (-1/2,1/2);
{{{abs(1-4x^2)=4x^2-1}}} everywhere else<br>
{{{11x-12-2x^2=-(2x^2-11x+12) = -1(2x-3)(x-4)}}}<br>
The behavior changes at x=3/2 and x=4.<br>
{{{abs(-(2x^2-11x+12))=-(2x^2-11x+12)}}} on the interval (3/2,4);
{{{abs(-(2x^2-11x+12))=(2x^2-11x+12)}}} everywhere else<br>
The critical points of the complete function divide the x-axis into 5 intervals on which the analysis needs to be done separately.<br>
(1) (-infinity,-1/2)
(2) (-1/2,1/2)
(3) (1/2,3/2)
(4) (3/2,4)
(5) (4,infinity)<br>
On intervals (1), (3), and (5),<br>
{{{abs(1-4x^2)-abs(11x-12-2x^2)=(4x^2-1)-(2x^2-11x+12)=2x^2+11x-13}}}<br>
and the inequality is<br>
{{{2x^2+11x-13<=1}}}
{{{2x^2+11x-14<=0}}}<br>
The quadratic formula gives the zeros of that quadratic expression as<br>
{{{(-11-sqrt(233))/4}}} = -6.566084
and
{{{(-11+sqrt(233))/4}}} = 1.066084<br>
Note one of those solutions is in interval (1) and the other is in interval (3), so both are valid solutions; and those solutions agree with what we found with a graphing calculator.<br>
I'll let you do the details if you want; but on intervals (2) and (4) there are no real solutions, so the results we have found are the complete answer.<br>
ANSWER: the solution set to the inequality is from {{{(-11-sqrt(233))/4}}} to {{{(-11+sqrt(233))/4}}}<br>