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#    |x^2-3*x-3| < |x^2+5*x-8|
f1 = function(x) abs(x^2-3*x-3)
f2 = function(x) abs(x^2+5*x-8)
# I understand that for x very large in absolute value the two functions behave in a
# similar way, such as x^2. Let's see the charts anyway.
graphF( f1, -50,50, "red"); graph ( f2, -50,50, "seagreen")
# I zoom around the origin:
graphF( f1, -4,5, "red"); graph ( f2, -4,5, "seagreen")
       
Diseq(f1,f2, -4,5, "blue")
x1 = more( solution2(f1,f2, -4,-2) )
# [1] -2.89791576165636
x2 = more( solution2(f1,f2, 0,1) )
# [1] 0.625
x3 = more( solution2(f1,f2, 1,3) )
# [1] 1.89791576165636
POINT(x1,f1(x1), "black")
POINT(x2,f1(x2), "black")
POINT(x3,f1(x3), "black")
#
I can try to write these approximate values in exact form using WolframAlpha.
If I put -2.89791576165636 and then put 1.89791576165636 I get:
1/2*(-1-sqrt(23))  and  1/2*(sqrt(23)-1)
I can also ask WolframAlpha directly for the resolution. With
solve abs(x^2-3*x-3) < abs(x^2+5*x-8) for x real      I have:
1/2*(-1-sqrt(23)) < x < 5/8  or  x > 1/2*(sqrt(23)-1)    OK