Finding all solutions to $cx \leq |x|$

Question

I'm looking find all $c \in \mathbb{R}$ that solve the inequality:

$cx \leq |x|$, for all $x \in \mathbb{R}$

I know the solution should be $-1 \leq c \leq 1$, but don't know how I am supposed to get there.

Could someone walk me through how to navigate the absolute value when there is an x outside the absolute value?

Answer 1

The equation you want to solve is equivalent to the system of the two equations : $$\forall x > 0, cx \leq x \quad \text{ and } \quad \forall x < 0, cx \leq -x $$

Obviously, the solutions of the first equation are the all the $c \leq 1$, and the solutions of the second one are all the $c \geq -1$.

So, the solutions of your equation are all the $-1 \leq c\leq 1$.

Answer 2

There are two cases to consider: $x>0$ and $x<0$.
If $x<0$, then $|x|=-x$.
So $cx\leq|x|\rightarrow c\geq\frac{|x|}{x}=\frac{-x}{x}=-1$

Now consider what happens for $x>0$, see if you can come to the solution $c\leq1$