+118654 k = -4 |k - 2| x |3k| -1
First I will do it the super long way.
I am going to split this into bits
k>=2
0<x<2
k<=0
\(If \;k\ge2 \;then\\ k = -4 |k - 2| x *|3k| -1\\ k=-4(k-2)*3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ k=\frac{-23\pm \sqrt{23^2-4*-12*-1}}{2*-12}\\ k=\frac{-23\pm \sqrt{528-48}}{-24}\\ k=\frac{-23\pm \sqrt{480}}{-24}\\ k=\frac{-23\pm 4\sqrt{30}}{-24}\\\)
\(k\approx\frac{-23\pm 21.9}{-24}\\ \text{neither of these answers are greater or equal to 2 so no solution here}\)
If 0<x<2
\(k = -4 |k - 2| * |3k| -1\\ k = -4*-(k - 2) * 3k -1\\ k = 12k(k - 2) -1\\ k = 12k^2-24k -1\\ 0 = 12k^2-25k -1\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k \approx \frac{25\pm25.9}{24}\\ \text{Neither of these answers are between 0 and 2. No solutions here either.}\)
\(If \;\; k\le0 \;\;then\\ k=-4|k-2|*|3k|-1\\ k=4(k-2)*-3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ 0=12k^2-23k+1\\ k=\frac{23\pm\sqrt{529-48}}{24}\\ k=\frac{23\pm\sqrt{481}}{24}\\ k\approx\frac{23\pm21.9}{24}\\ \mbox{No solution here either }\)
No solutions 
+118654 k = -4 |k - 2| x |3k| -1
First I will do it the super long way.
I am going to split this into bits
k>=2
0<x<2
k<=0
\(If \;k\ge2 \;then\\ k = -4 |k - 2| x *|3k| -1\\ k=-4(k-2)*3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ k=\frac{-23\pm \sqrt{23^2-4*-12*-1}}{2*-12}\\ k=\frac{-23\pm \sqrt{528-48}}{-24}\\ k=\frac{-23\pm \sqrt{480}}{-24}\\ k=\frac{-23\pm 4\sqrt{30}}{-24}\\\)
\(k\approx\frac{-23\pm 21.9}{-24}\\ \text{neither of these answers are greater or equal to 2 so no solution here}\)
If 0<x<2
\(k = -4 |k - 2| * |3k| -1\\ k = -4*-(k - 2) * 3k -1\\ k = 12k(k - 2) -1\\ k = 12k^2-24k -1\\ 0 = 12k^2-25k -1\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k \approx \frac{25\pm25.9}{24}\\ \text{Neither of these answers are between 0 and 2. No solutions here either.}\)
\(If \;\; k\le0 \;\;then\\ k=-4|k-2|*|3k|-1\\ k=4(k-2)*-3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ 0=12k^2-23k+1\\ k=\frac{23\pm\sqrt{529-48}}{24}\\ k=\frac{23\pm\sqrt{481}}{24}\\ k\approx\frac{23\pm21.9}{24}\\ \mbox{No solution here either }\)
No solutions
+118654 Lets look at it a different way.
k = -4 |k - 2| x |3k| -1
\(k = -4 |k - 2| x* |3k| -1\)
3k and (k-2) are both positve when k>2
3k and (k-2) are both negative when k<0
so for 0<k<2 the equation becomes
\(k = -4 (k - 2) (3k) -1\\ k = -12k (k - 2) -1\\ k = -12k^2 +24k -1\\ 0= -12k^2 +23k -1\\ \text{Consider the parabola}\\ y= -12k^2 +23k -1\\ \)
This is a concave down parabola
axis of sym y= -23/-24 = 23/24 this is between 0 and 2
when k=0 y=-1
when k=2 y = -48+46-1<0
So no solutions here.
Between k=0 and k=2
3k >=0 and (k-2) <=0
so the equation becomes
\(k = 4 (k - 2) (3k) -1\\ k = 12k (k - 2) -1\\ k = 12k^2 -24k -1\\ 0= 12k^2 -25k -1\\ \text{Consider the concave up parabola}\\ y= 12k^2 -25k -1\\ \text{axis of symmetry=25/24 and this is between 0 and 2}\\ \text{When k=0 y=-1, when k=2 y=48-50-1=-3}\\ \text{this means that between k= 0 and k=2 the y value is always negative }\\\)
No solutions here either.
Here is the graph
https://www.desmos.com/calculator/moids6hkw3
