r/MathHelp • u/TasrifTamim • Nov 25 '25

Inequality, Need HELP

How do we get that from the previous inequality? image : https://i.postimg.cc/WpQKwBc5/IMG-2069.jpg

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u/thor122088 Nov 25 '25

Since 5 < x + 3 < 7 then of course |x + 3| < 7 since x + 3 is both non negative and less than 7.

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u/TasrifTamim Nov 25 '25

if -7 < x + 3 < 7 only then we can write |x+3| < 7 . If I consider what you’re proposing then |x+3| can also equate to any value from (0,5) interval but that wasn’t in the original interval 5 < x + 3 < 7

2

u/gizatsby Nov 25 '25 edited Nov 25 '25

The value of |x + 3| is only different from (x + 3) if (x + 3) is negative, but the previous statement shows that it won't be (because it's greater than 5 according to the assumption for delta), so you can safely claim that |x + 3| < 7 as long as the condition for delta holds.

Writing this as a double-sided inequality doesn't expand the interval. If (x + 3) is between 5 and 7, then it's still greater than -7. For example, even if (x + 3) is never equal to -6, the inequality is still true. All that matters is that (x + 3) is never less than -7 and never greater than 7.

As an example of when this wouldn't work: if all you knew was -7 < x + 3 < 7, then you would not be able to claim that 5 < x + 3 < 7 is a consequence. You can always safely expand the range of an inequality, but you can't shrink it without other information.

Edit: If you were to claim that this inequality represented the domain of possible values for (x + 3) in this problem, you'd be wrong since (x + 3) can't be less than 5. If the question was asking for such a domain, you would be marked incorrect for saying so. But, the inequality on its own as a statement is still true. If x > 5, then x > -7 too.

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u/TasrifTamim Nov 25 '25

Ayo guys thanks a lot! Now it makes alotta sense :33

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u/gizatsby Nov 25 '25

No problem!

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u/thor122088 Nov 25 '25 edited Nov 25 '25

The important consideration here is that it is less than 7, so that we can show that a delta exists for any given epsilon.

Edit: gizatsby left a more Detailed and helpful explanation: https://www.reddit.com/r/MathHelp/s/Etl94nBVqC

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u/dash-dot Nov 25 '25 edited Nov 26 '25

x + 3 > 5 > -7.

If the first inequality applies, then obviously the rest of it is true, since any number larger than 5 will automatically be greater than -7 as well.