Question

Rezolvati ecuatiile :
1. |5-|2x+1||= 8
2. ||x-2|-4|=6
3. 8√2x-3(5√2x-4)+7√2=2(6x-4√2)+2√2(x+3√2)

Answer 1

$\bold{1.}~ \Rightarrow 5-|2x+1|=8~sau~5-|2x+1|=-8. \\ \\ 5-|2x+1|=8 \Rightarrow |2x+1| =-3,~imposibil. \\ 2-|2x+1|=-8 \Rightarrow |2x+1|=6 \Rightarrow 2x+1=6~sau~2x+1=-6. \\ 2x+1=6 \Rightarrow x= \frac{5}{2} . \\ 2x+1=-6 \Rightarrow x= -\frac{7}{2}. \\ \\ \underline{Solutie}: x \in \{- \frac{7}{2}, \frac{5}{2} \}.$

 $\bold{2.}~ \Rightarrow |x-2|-4=6~sau~|x-2|-4=-6. \\ \\ |x-2|-4=6 \Rightarrow |x-2|=10 \Rightarrow x-2=10~sau~x-2=-10 \Rightarrow \\ \Rightarrow x \in \{-8;12\}. \\ \\ |x-2|-4=-6 \Rightarrow |x-2|=-2,~imposibil. \\ \\ \underline{Solutie}: x \in \{-8;12\}.$

$8 \sqrt{2}x-3(5 \sqrt{2}x-4)+7 \sqrt{2}=2(6x-4 \sqrt{2})+2 \sqrt{2}(x+3 \sqrt{2}) \\ 8 \sqrt{2}x-15 \sqrt{2}x+12+7 \sqrt{2}=12x-8 \sqrt{2}+2 \sqrt{2}x+12 \\ 8 \sqrt{2}x-15 \sqrt{2}x-12x-2 \sqrt{2}x=-8 \sqrt{2}+12-12- 7\sqrt{2} \\ x( 8\sqrt{2} -15 \sqrt{2}-12-2 \sqrt{2})=-15 \sqrt{2} \\ x(-9 \sqrt{2} -12)=-15 \sqrt{2} \\ x= \frac{-15 \sqrt{2} }{- 9\sqrt{2}-12 } \\ x= \frac{15 \sqrt{2} }{9 \sqrt{2} +12} \\ x= \frac{15 \sqrt{2}(9 \sqrt{2}-12) }{18}$
$x= \frac{5 \sqrt{2}(9 \sqrt{2}-12 ) }{6} .$