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Solve this equation.

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2 Answers

1) If x+5 ≥ 0, then x+5 ≥ x is always true since 5 ≥ 0. So, x ≥ -5 is a solution.
Or if x+5 < 0, then -x-5 > x, which leads to x < -5/2.
Combining the two solutions, you get the answer - ∞ < x < ∞
Answer: - ∞ < x < ∞
 
 
2) Critical points: x = -1, 2.5
In (-∞, -1], -(x+1)-2 = -(2x-5). =>x = 8. So, there is no solution.
In (-1, 2.5], x+1-2 = -(2x-5). => x = 2 (solution 1.)
In (2.5, ∞), x+1-2 = 2x-5. => x = 4 (solution 2.)
Answer: x = 2, 4
x+5 is always greater than x and |x+5|≥x+5.  - ∞ < x < ∞ is answer part 1
 
part 2. | x + 1 | -2 =| 2x - 5 | or | x + 1 | = 2+ | 2x - 5 |.  This problem is very well seen by drawing a picture.
Draw y=| x + 1 | and y= 2+ | 2x - 5 | on the same graph.  The only intersections are the two where
y=x+1 and the y= 2+ |2x - 5| branches.
a.  y=x+1 and y= 2+2x-5 or y=x+1 and y=2x-3, that is x+1=2x-3 or 4=x
b. y=x+1 and y= 2-2x+5  or y=x+1 and y=-2x+7, that is x+1=-2x+7 or 3x=6 or x=2