How do you solve the inequality #x^2>=absx# and write your answer in interval notation?

1 Answer
Oct 4, 2017

Answer:

#x in (-oo,-1]uu(+1,+oo)#

Explanation:

There are several ways to solve this:

Algebraically

If #x^2 >= abs(x)#
then we can examine the two cases:

#{: ("if "x >=0,x^2>=abs(x)rarr,x^2 >=x,color(white)("xxxx"),"if "x <0,x^2 >=abs(x)rarr,x^2>=-x), (,,,,x >=1,,-x >= 1), (,,,,,,x <=1) :}#

Graphically
Compare the graphs of #x^2# and #abs(x)# for the ranges where #x^2>=abs(x)#
enter image source here

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Since the solution is a union of two ranges we use the #uu# symbol to connect our specification of each range.

The range for #x < 0#
extends to #-oo# ...but this is not a real value, only a limit, so we use the "open" parenthesis to show that it is not really included: #(-oo#
and
includes #-1# ...where #-1# is part of the solution, so we use the "closed" parenthesis to show that it is included: #-1]#

Similarly for the # x >= 0# range notation.