3
$\begingroup$

I've just started with absolute value equations and I have a real hard time understanding how to solve this. I got the following question, and I can't make heads or tails out of it.

Assume that $x, y$ are points on the real line. Explain what $|x − y|$ means geometrically. Use this to illustrate the inequalities (as subsets of the real line) below and write the inequality without absolute values.

$3 > |x + 4| \geq 1$

All help will be much appreciated, even as much as small pointers will be more than welcome...

$\endgroup$
  • $\begingroup$ Ok, If you have just started with absolute values, is this the first equation you were confronted with?? $\endgroup$ – imranfat Sep 18 '15 at 22:39
  • $\begingroup$ In this particular setup yes, how would one illustrate the inequalities is what I have a hard time to comprehend. $\endgroup$ – Breezer Sep 18 '15 at 22:44
  • $\begingroup$ Apart from the other suggestions below, do you know how to graph the function$y=|x+4|$ in the $xy$ plane? $\endgroup$ – imranfat Sep 19 '15 at 3:50
5
$\begingroup$

$|x-y|$ is the distance between $x$ and $y$ on a number line.

$\endgroup$
4
$\begingroup$

Building up on what Aleksandar already said. Your inequality can be written as:$$1\le|x-(-4)|<3$$ So geometrically, $x$ are the numbers on the real line such that they are at least a distance 1 from -4 and at most (but not equal to) a distance 3 from -4. So $$-7<x\le-5$$ or $$-3\le x<-1$$ A more systematic way of doing such questions is to consider the 2 possibilities i.e $1\le x+4<3$ and $1\le -(x+4)<3$ and simplying should give the same answer as above.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.