1. Let $$f(x) =2x-5$$ and $$g(x) =7-2x$$. Then |f(x)+ g(x)| = |f(x)|+ |g(x)| if and only if
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By: anil on 05 May 2019 02.29 am
$$|f(x)+ g(x)| = |f(x)| + |g(x)|$$ if and only if
case 1: $$f(x) geq 0$$ and $$g(x) geq 0$$
$$ 2x-5 geq 0 $$ and $$7-2x geq 0$$
$$ x geq frac{5}{2}$$ and $$ frac{7}{2} geq x$$
$$frac{5}{2}leq xleqfrac{7}{2}$$
case 2: $$f(x) leq 0$$ and $$g(x) leq 0$$ $$ 2x-5 leq 0 $$ and $$7-2x leq 0$$
$$ x leq frac{5}{2}$$ and $$ frac{7}{2} leq x$$
So x=7/2 which is not possible. Hence, answer is $$frac{5}{2}leq xleqfrac{7}{2}$$
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case 1: $$f(x) geq 0$$ and $$g(x) geq 0$$
$$ 2x-5 geq 0 $$ and $$7-2x geq 0$$
$$ x geq frac{5}{2}$$ and $$ frac{7}{2} geq x$$
$$frac{5}{2}leq xleqfrac{7}{2}$$
case 2: $$f(x) leq 0$$ and $$g(x) leq 0$$ $$ 2x-5 leq 0 $$ and $$7-2x leq 0$$
$$ x leq frac{5}{2}$$ and $$ frac{7}{2} leq x$$
So x=7/2 which is not possible. Hence, answer is $$frac{5}{2}leq xleqfrac{7}{2}$$