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

Oct 4, 2017

$x \in \left(- \infty , - 1\right] \cup \left(+ 1 , + \infty\right)$

Explanation:

There are several ways to solve this:

Algebraically

If ${x}^{2} \ge \left\mid x \right\mid$
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 $\left\mid x \right\mid$ for the ranges where ${x}^{2} \ge \left\mid x \right\mid$

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Since the solution is a union of two ranges we use the $\cup$ symbol to connect our specification of each range.

The range for $x < 0$
extends to $- \infty$ ...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 $x \ge 0$ range notation.