Inequalities help

Bogogogo

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I’m confused on how to work inequalities. Here is a problem on my homework that I’m trying to figure out.


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topsquark

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Please let us know what you have tried. We can help you better that way.

-Dan

Hint: 13 > x > 8 implies that 13 > 8, so that can't be right. What are the possible values of x within 5 units of 13 where x > 13?
 

pka

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I’m confused on how to work inequalities. Here is a problem on my homework that I’m trying to figure out.


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The expression \(\displaystyle |x-13|\) is read "the distance from \(\displaystyle x\text{ to }13\)".
So what is the inequality?
 

topsquark

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Hint: 13 > x > 8 implies that 13 > 8, so that can't be right. What are the possible values of x within 5 units of 13 where x > 13?
Addendum: I wrote the inequalities backward. I should have written 13 < x < 8 , as you wrote, leading to 13 < 8.

-Dan
 

Bogogogo

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The expression \(\displaystyle |x-13|\) is read "the distance from \(\displaystyle x\text{ to }13\)".
So what is the inequality?
Would it be I x-3 I< 5
 

JeffM

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Would it be I x-3 I< 5
I think you meant \(\displaystyle |x - 13| < 5.\)

That answer is wrong as you can tell by letting x = 13. Obviously 13 is not at least 5 away from itself.

I like to think about inequalities by first thinking about the equivalent equality.

\(\displaystyle x - 13 = 5 \implies x = 18 \implies x > 18 \implies x - 13 > 5.\)

\(\displaystyle 13 - x = 5 \implies WHAT?\)
 

pka

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Would it be I x-3 I< 5
The correct answer is \(\displaystyle |x-13|\ge 5\) OR \(\displaystyle \{x | x\in (-\infty,8]\cup [18,\infty)\}\)
 

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Bogogogo

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I think you meant \(\displaystyle |x - 13| < 5.\)

That answer is wrong as you can tell by letting x = 13. Obviously 13 is not at least 5 away from itself.

I like to think about inequalities by first thinking about the equivalent equality.

\(\displaystyle x - 13 = 5 \implies x = 18 \implies x > 18 \implies x - 13 > 5.\)

\(\displaystyle 13 - x = 5 \implies WHAT?\)
So that would be 13-x=5, x=8, x>8, 13-x>5
 

JeffM

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You are guessing.

So if a number x is at least 5 from 13, then either

\(\displaystyle 13 - x \ge 5 \text { or } x - 13 \ge 5.\)

Do you follow that?

Now what I recommend, for any kind of inequality, is to solve the corresponding equality. So let's think about the first inequality.

\(\displaystyle 13 - x \ge 5 \iff 13 - x = 5 \text { or } 13 - x > 5.\)

The first is obvious \(\displaystyle 13 - x = 5 \implies 13 - 5 = x \implies x = 8.\)

Then

\(\displaystyle 13 - x > 5 \implies 13 - 5 > x \implies 8 > x.\)

Perfectly similar to how you solved the equality. Now put them together.

\(\displaystyle x = 8 \text { or } 8 > x \implies 8 \ge x \implies x \le 8.\)

But this is only half the problem.

Now work through \(\displaystyle x - 13 \ge 5.\)
 
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