Solve this equation.

Question

Solve:

 

1.) |x+5| > x

2.) | x + 1 | -2 =| 2x - 5 |

Answer 1

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

 

Answer 2

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