I suppose you don't like problem like this: \(|x-3|+3*|x-5|=|x+4|\) What is x?
I think there is one general and right approach based on a general rule:
\(|x|=\left{ \\
\begin{eqnarray} \\
x: x\ge0 \\\\
-x: x<0 \\\\
\end{eqnarray}\)
Now we try to solve our horrible problem.
1. the first step1.1 conditions: \((x-3)\ge0\); \((x-5)\ge0\); \((x+4)\ge0\)
x e \([3;+\infty)&[5;+\infty)&[-4;+\infty)=[5;+\infty)\)
1.2 solution:
\((x-3)+3*(x-5)=(x+4)\)
\((x-3)+3*(x-5)-(x+4)=0\)
\(3x-14=0\)
\(x=4\frac{2}{3}\)
2.3 check conditions:
\(x=4\frac{2}{3}\) is out of \([5;+\infty)\)
no solutions2. the second step2.1 conditions: \((x-3)<\); \((x-5)\ge0\); \((x+4)\ge0\)
x e \((-\infty;3)&[5;+\infty)&[-4;+\infty)=(empty)\)
no solutions3. the third step3.1 conditions: \((x-3)\ge0\); \((x-5)<0\); \((x+4)\ge0\)
x e \([3;+\infty)&(-\infty;5]&[-4;+\infty)=[3;5]\)
3.2 solution:
\((x-3)-3*(x-5)=(x+4)\)
\((x-3)-3*(x-5)-(x+4)=0\)
\(-3x-16=0\)
\(x=-5\frac{1}{3}\)
3.3 check conditions:
\(x=-5\frac{1}{3}\) is out of \([3;5]\)
no solutions4. the fourth step4.1 conditions: \((x-3)\ge0\); \((x-5)\ge0\); \((x+4)<0\)
x e \([3;+\infty)&[5;+\infty)&(-\infty;-4)=(empty)\)
no solutions5. the fifth step5.1 conditions: \((x-3)<\); \((x-5)<0\); \((x+4)\ge0\)
x e \((-\infty;3]&(-\infty;5]&[-4;+\infty)=[-4;3]\)
5.2 solution:
\(-(x-3)-3*(x-5)=(x+4)\)
\(-(x-3)-3*(x-5)-(x+4)=0\)
\(-5x+14=0\)
\(x=2\frac{4}{5}\)
5.3 check conditions:
\(x=2\frac{4}{5}\) is in \([-4;3]\)
solution is \(x=2\frac{4}{5}\)6. the sixth step6.1 conditions: \((x-3)<\); \((x-5)\ge0\); \((x+4)<0\)
x e \((-\infty;3]&[5;+\infty]&[-\infty;-4)=(empty)\)
no solutions7. the seventh step7.1 conditions: \((x-3)<\); \((x-5)\ge0\); \((x+4)<0\)
x e \([3;+\infty]&(-\infty;5]&[-\infty;-4)=(empty)\)
no solutions8. the eighth step8.1 conditions: \((x-3)\ge0\); \((x-5)\ge0\); \((x+4)\ge0\)
x e \((-\infty;3]&(-\infty;5]&[-\infty;-4)=(-\infty;-4]\)
8.2 solution:
\(-(x-3)-3*(x-5)=-(x+4)\)
\(-(x-3)-3*(x-5)+(x+4)=0\)
\(-3x+22=0\)
\(x=7\frac{1}{3}\)
8.3 check conditions:
\(x=7\frac{1}{3}\) is out of \([-\infty;-4]\)
no solutions____________________________________________________________
There is only one solution: \(x=2\frac{4}{5}\)I've try to write solution step by step. I real word you can go through condition step very fast.
Hope this help.
I hope there is no typo