Does this inequality make sense? 1 = |1|?

Question

Okay so suppose x = -9.

Then we have x < 1 .

But 1 = |1| Hence x < |1| Implies -1 < x < 1

But this clearly is not true. Just wondering what the limitations are when using inequalities or is there a limitation with the absolute value. Or am I just missing something?

Also I wondered. If x< 4 and x > 4, can we say x = 4?

Answer 1

You are confusing $|x| < 1$ with $x < |1|$.

when $x=-9$, the first inequality that I have written above is not true.

Also, there is no number that satisfies $x<4$ and $x > 4$.

However, if $x \le 4$ and $x \ge 4$, then we can say that $x=4$.

Answer 2

You seem to be assuming that if $a<b$ then $|a|<|b|$, but that is not true. More generally, if $a<b$ that does not necessarily imply $f(a)<f(b)$ for any function $f$. However, it is true if $f$ is strictly increasing (which the absolute value function isn't).

If $x<|1|$ that is the same as saying $x<1$ because the absolute value of 1 is simply 1. However, If $|x|<1$ then it is true that $-1<x<1$.

If $x<4$ and $x>4$ then $x$ does not exist, because no real number satisfies both of those conditions.

Answer 3

There is some confusion here. It is not true that $x<\lvert1\rvert\implies-1<x<1$. Perhaps that you are thinking about $\lvert x\rvert<1\implies-1<x<1$.

And if $x<4$ and $x>4$, then there is no such $x$. In particular, you can't say that $x=4$.