How do you solve #abs(5-x)<=x#?

1 Answer
Douglas K.
Oct 4, 2017

Use the piecewise definition of the absolute value function, #|f(x)|={(f(x);f(x)>=0),(-f(x);f(x)<0):}# to separate into two inequalities.
Solve both inequalities.

Explanation:

Separate into two inequalities:

#5-x <=x# and #-(5-x) <= x#

Distribute the minus sign in the second inequality:

#5-x <=x# and #-5+x <= x#

Add x to both sides of the first inequality and subtract x from both sides of the second:

#5 <=2x# and #-5 <= 0#

The second inequality does not depend on x, therefore, we discard it.

Divide both sides of the first inequality by 2:

#5/2 <=x#

Flip the inequality

#x >= 5/2#

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