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45 votes
|x+1|+|x-2|=3
WHAT IS THE ANSWER??

User Igor Maznitsa
by
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1 Answer

14 votes
14 votes

Answer:

-1 ≤ x ≤ 2

Explanation:

The given equation can be rewritten as a piecewise function and solved in pieces.

The term |x+1| changes definition at x=-1. For x-values less than that, it is defined as -(x+1). Similarly, the term |x-2| changes definition at x=2. For x-values less than that, its definition is -(x-2). In the following, we consider the three pieces of the function.

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x < -1

In this domain, the absolute value functions both have negative arguments, so the equation is effectively ...

-(x +1) -(x -2) = 3

-2x +1 = 3 . . . . . . . simplify

-2x = 2 . . . . . . subtract 1

x = -1 . . . . divide by -2

This value of x is not in the domain we have defined, so there is no solution in this domain.

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-1 ≤ x ≤ 2

In this domain, the function |x+1| has a non-negative argument, but the function |x-2| has a non-positive argument. The equation is effectively ...

(x +1) -(x -2) = 3

x -x +1 +2 = 3 . . . . . eliminate parentheses, group like terms

3 = 3 . . . . . . . . . true for the entire domain

The solution here is -1 ≤ x ≤ 2.

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2 < x

Both absolute value terms have positive arguments in this domain, so the equation is effectlvely ...

(x +1) +(x -2) = 3

2x -1 = 3 . . . . . simplify

2x = 4 . . . . add 1

x = 2 . . . . divide by 2

This value of x is not in the domain we have defined, so there is no solution in this domain.

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The solution to the given equation is -1 ≤ x ≤ 2.

|x+1|+|x-2|=3 WHAT IS THE ANSWER??-example-1
User David Duponchel
by
3.0k points