How do I even go about solving something like this:
$|x-1| - |x-3| \geq 5$
I know that if the question is this:
$|x-1| > 5$
I can split that up into:
$|x-1| > 5$
and
$|x-1| < -5$
But I'm a little confused about this problem.
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$\begingroup$ You could use the same idea and split into $\,x-1 \ge 5 + |x-3|\,$ or $\,x-1 \le - 5 - |x-3|\,$. Then repeat. $\endgroup$– dxivCommented Jan 5, 2018 at 1:16
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$\begingroup$ Can someone confirm my suspicion? It's a weird question because I believe there are no real solutions to the equation. $\endgroup$– LandurosCommented Jan 5, 2018 at 1:20
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1$\begingroup$ @Landuros Right, and it follows directly from the triangle inequality. $\endgroup$– dxivCommented Jan 5, 2018 at 1:25
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$\begingroup$ I always have a hard time with this but $|a| + |b| \ge |a + b|$ so $|a| \ge |a+b| - |b|$ so let $a + b = c$ and $|c - b| \ge |c| - |b|$. .... So $|(x-1)-(x-3)| = 2 \ge |(x-1)| - |(x-3)|$. And so so $|x-1|-|x-3| \ge 5 > |(x-1)-(x-3)| = 2$ is impossible. If you can get used to this "triangle inequality with subtraction" is is very useful and powerful. But... I'm embarrassed to admit, I keep having trouble with it all these decades later. $\endgroup$– fleabloodCommented Jan 5, 2018 at 1:56
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$\begingroup$ Where does the definition of triangle inequality come from? I thought the triangle inequality was if you had two sides, say 4 and 6, the third side must be between 2 and 10. $\endgroup$– Jwan622Commented Jan 5, 2018 at 22:52
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2 Answers
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You are right to be suspicious:
$|x-1| - |x-3| \ge 5 \implies$
$|x - 1| \ge 5 + |x-3| \ge 5$ so
$x -1 \ge 5$ or $x - 1 \le -5$ so $x \ge 6$ or $x \le -4$.
But if $x \ge 6$ then $x > 3$ and $|x-1| = x-1$ and $|x-3| = x-3$ so
$(x-1) - (x-3) = 2 \ge 5$ which is clearly false.
And if $x \le -4$ then $x-1 < 0$ and $x - 3< 0$ so
$|x-1| - |x-3| = -(x-1)- (-(x-3)) = -x + 1 + x -3 = -3 \ge 5$ which is even more obviously false.
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Well, you can always do four choises:
1)$(x-1) \ge 0$ and $x-3 \ge 0$ so $x \ge 1$ and $x \ge 3$
So $x\ge 3$ and $|x-1| - |x-3| = (x-1) -(x-3) = -1+3 = 2 \ge 5$ That's impossible.
2)$x-1 < 0$ and $x \ge 0$ so $x < 1$ and $x \ge 3$.
That is impossible.
3) $x-1 \ge 0$ and $x - 3 < 0$ so $x \ge 1$ and $x < 3$ so $1 \le x <3$.
So $|x-1| - |x-3| = (x-1) - (3-x) = 2x -4 \ge 5$ so $2x \ge 9$ $x \ge 4.5$.
So $1 \le x < 3$ and $x \ge 4.5$.
That is ismpossible.
Or
4) $x-1 < 0$ and $x -3 < 0$ so $x < 1$ and $x < 3$ so $x < 1$
So $|x-1| - |x-3| = (1-x) - (3-x) = -2 \ge 5$.
That is impossible.
There is not solution.
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Then there is the triangle inequality..
$|a+b| \le |a| + |b|$ so
$|a + b| - |b| \le |a|$. Let $a+b= c$
$|c| - |b| \le |c-b|$
So $|x-1| - |x-3| \le |(x-1) -(x-3)| = |2| = 2$.
So $|x-1| - |x-3| \ge 5 > 2$ is impossible.
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You need to partition the real line into segments, such that for each segemnt the signs of $x-1$ and $x-3$ stay unchanged. For instance, when $x\in (\infty,1)$ the inequality becomes
$$-(x-1)-(-(x-3))\geq5$$
it can be simplified to
$$-2\geq5$$
which is never true.
So, you do the same for all the partitions $(a,b)$ and if you find some solutions for $x$, you take the intersection of the solution with $(a,b)$, because the simplification of the main inequality is only valid on $(a,b)$.
