Answer to Question #3878 in Real Analysis for junel

Question

Find all x ∈ R that satisfy the equation& |x+1| + |x-2| = 7.

Answer 1

|x+1|+|x-2|=7

Let’s estimate our x using the simple property of module:
& |x|& = { -x, x<0
& x, x>0
Imagine that we’re dividing our rational numbers into three groups: x ∈(-∞,-2),
x∈(-2,-1),& x∈(-1,+∞) we have such systems.
{x+1<0 & -x-1-x+2=7
& x-2<0) x = -3

{x+1>0 x+1-x+2=7
& x-2<0 x-∄

{x+1<0 -x-1+x-2=7
& x-2>0 & & x-∄

{x+1>0 x+1+x-2=7
& x-2>0 & x = 4

Answer: x=-3, x=4