Answer:
[tex]y \ge9 [/tex]
Step-by-step explanation:
The given function is
[tex]y = |x - 3| + |x + 2| - |x - 5| [/tex]
By the triangle inequality property,
[tex] |x + y| \leqslant |x| + |y| [/tex]
and the alternate triangle inequality property;
[tex] |x - y| \geqslant |x| - |y| [/tex]
We apply this property to get:
[tex]y \geqslant |x | - |3| + |x | + |2| -( |x | - |5| )[/tex]
Expand parenthesis to get:
[tex]y \geqslant |x | - |3| + |x | + |2| - |x | + |5|[/tex]
Simplify to get:
[tex]y \geqslant |x | - 3+ |x | + 2- |x | + 5[/tex]
Group similar terms to get:
[tex]y \geqslant |x | + |x | - |x | - 3 + 2+ 5[/tex]
[tex]y \geqslant |x | + 4 [/tex]
When x>5,
[tex]y \geqslant |5 | + 4 [/tex]
That's;
[tex]y \geqslant 5 + 4 \\ y \geqslant 9[/tex]
