Answer:
Step-by-step explantion:
Definition of absolute value of x:
|x|=x assuming x has taken a positive value.
|x|=-x assuming x has taken a negative value.
|x|=0 assuming x is 0.
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[tex]-x \le |x|[/tex]
Add [tex]x[/tex] on both sides:
[tex]0 \le |x|+x[/tex]
[tex]|x|+x \ge 0[/tex]
Let [tex]h(x)=|x|+x[/tex].
Assume x is positive, then [tex]h(x)=x+x=2x[/tex] where [tex]2x[/tex] is positive since [tex]x[/tex] is positive. Also [tex]h(x)=2x>0[/tex] for positive x.
Assume [tex]x[/tex] is negative, then [tex]h(x)=-x+x=0[/tex].
Assume [tex]x=0[/tex], then [tex]h(0)=|0|+0=0+0=0[/tex].
The result for h is positive when x is positive but 0 elsewhere for x.
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