\(1- \frac{6(x+3)}{|4+2x|}>-1 \)
\(- \frac{3(x+3)}{|2+x|}>-2|\cdot(-1) \)
\(\frac{3(x+3)}{|2+x|}<2|^{2} \)
\(\frac{9(x+3)^2}{(2+x)^2}<4\)
\(\frac{(x+3)^2}{(2+x)^2}<\frac{4}{9}\)
\((\frac{x+3}{2+x})^2<\frac{4}{9}|±\sqrt{~~}\)
1.)
\(\frac{x+3}{2+x}<\frac{2}{3}\)
\(x+3<\frac{2}{3}(2+x)\)
\(x+3<\frac{4}{3}+\frac{2}{3}x)\)
\( \frac{1}{3}x<-\frac{5}{3} \)
\( x_1<-5 \)
Probe mit \(x=-6\)
\(1- \frac{6\cdot(-6+3)}{|4-12|}>-1 \)✓
2.)
\(\frac{x+3}{2+x}<-\frac{2}{3}\)
\(x+3<-\frac{2}{3}\cdot(2+x)\)
\(x+3<-\frac{4}{3}-\frac{2}{3}x)\)
\(x+3<-\frac{4}{3}-\frac{2}{3}x\)
\(\frac{5}{3}x<-\frac{13}{3}\)
\(x_2<-\frac{13}{5}\)
Probe mit \(x=-3\):
\(1- \frac{6\cdot (-3+3)}{|4-10|}>-1 \)✓
\((-∞,-\frac{13}{5})\)
Proben sind notwendig, weil Quadrieren keine Äquivalenzumformung ist.