Inequality with two absolute values
I'm new here, and I was wondering if any of you could help me out with this little problem that is already getting on my nerves since I've been trying to solve it for hours.
Studying for my next test on inequalities with absolute values, I found this one:
$$ |x-3|-|x-4|<x $$ (I precisely found the above inequality on this website, here to be precise, but, the problem is that when I try to solve it, my answer won't be $(-1,+\infty)$, but $(1,7)$. I took the same inequalities that the asker's teacher had given to him and then put them on a number line, and my result was definitely not $(-1,+\infty)$
Here are the inequalities: $$ x−3 < x−4 +x $$ $$ x−3 < −(x−4) +x $$ $$ −(x−3)<−(x−4)+x $$
And here are my answers respectively: $$ x>1, \quad x>-1, \quad x<7 $$
I will really appreciate if anyone could help me out, because I'm already stressed dealing with this problem, that by the way, it is not demanding that I solve it, but you know, why not?
How to solve $|x-3|-|x-4|<x$? Let $f(x)=|x-3|-|x-4|$
I would begin by noting $|x-3|=x-3$ for $x \geq 3$ whereas $|x-3|=3-x$ for $x \leq 3$. Likewise, $|x-4|=x-4$ for $x \geq 4$ whereas $|x-4|=4-x$ for $x \leq 4$
if $x \geq 4$ then $x > 3$ hence $f(x)=x-3-(x-4)=1$ and we find $1<x$ which is true in this case. This puts $[4,\infty)$ in the solution set of the inequality.
if $x \leq 3$ then $x < 4$ hence $f(x)=3-x-(4-x)=-1$ and we face $-1<x$ which is true for each $x$ with $-1<x\leq 3$. This shows $(-1,3]$ is also in the solution set of the inequality.
if $3<x<4$ then $f(x) = x-3-(4-x)=2x-7$. Hence in this context we must solve $2x-7<x$ which gives $x<7$ which is true for each $x$ in the interval considered. This places $(3,4)$ in the solution set of the inequality.
Put all of this together, we obtain $(-1,3] \cup (3,4) \cup [4,\infty) = (-1, \infty)$ which is your answer. Usually I solve this sort of thing with a sign-chart method.
What you have is almost correct, the last step is to restrict your solution to the corresponding region.
For example, for $x>1$, the answer must be in the $x>4$ region, so your answer for this region is $x>4$.
For $x>-1$, your answer must be in $x<3$ region, so your answer for this region is $-1<x<3$.
And for the last one, $x<7$, your answer must be in the corresponding region we had first, namely $3<x<4$, so your answer for this region is $3<x<4$.
Now draw these three answers on the number line and you'll have $x>-1$, the desired final answer.
The standard semi-mechanical way to eliminate the absolute value signs is to divide the number line into segments. The critical point for $|x-4|$ is at $x=4$, and the critical point for $|x-3|$ is at $x=3$.
Suppose first that $x \ge 4$. Then $|x-4|=x-4$ and $|x-3|=x-3$. So we are looking at the inequality $(x-3)-(x-4)\lt x$, that is, at $1\lt x$, which is certainly true for $x\ge 4$.
Now suppose that $3\le x\lt 4$. Then $|x-4|=4-x$ and $|x=3|=x-3$, so we are looking at the inequality $(x-3)-(4-x)\lt x$, that is, $2x-7\lt x$. This simplifies to $x\lt 7$, which is certainly true in the interval $[3,4)$.
It is probably at this point that your calculation went astray. We were looking at the interval $[3,4)$ and asking which points in this interval satisfied our inequality. The manipulation told us that it was all points in this interval that satisfied $x\lt 7$. Well, they all do!
Finally, suppose that $x\lt 3$. Then $|x-4|=4-x$ and $|x-3|=3-x$. So we are looking at the inequality $(3-x)-(4-x)\lt x$. Calculate. The left side is $-1$, so for in the interval $(-\infty,3)$, the inequality holds precisely when $-1\lt x$.
Putting things together, we conclude that the original inequality holds (i) if $x\ge 4$; (ii) if $3\le x\lt 4$; and (ii) if $x\lt 3$ but $-1\lt x$. This complicated set of conditions can be summarized much more simply as $x\gt -1$.
There are other ways to describe the set of solutions. For example we could say that the set is $(-1,\infty)$.