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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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?
 
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
 
Would it be I x-3 I< 5
I think you meant [MATH]|x - 13| < 5.[/MATH]
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.

[MATH]x - 13 = 5 \implies x = 18 \implies x > 18 \implies x - 13 > 5.[/MATH]
[MATH]13 - x = 5 \implies WHAT?[/MATH]
 
I think you meant [MATH]|x - 13| < 5.[/MATH]
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.

[MATH]x - 13 = 5 \implies x = 18 \implies x > 18 \implies x - 13 > 5.[/MATH]
[MATH]13 - x = 5 \implies WHAT?[/MATH]
So that would be 13-x=5, x=8, x>8, 13-x>5
 
You are guessing.

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

[MATH]13 - x \ge 5 \text { or } x - 13 \ge 5.[/MATH]
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.

[MATH]13 - x \ge 5 \iff 13 - x = 5 \text { or } 13 - x > 5.[/MATH]
The first is obvious [MATH]13 - x = 5 \implies 13 - 5 = x \implies x = 8.[/MATH]
Then

[MATH]13 - x > 5 \implies 13 - 5 > x \implies 8 > x.[/MATH]
Perfectly similar to how you solved the equality. Now put them together.

[MATH]x = 8 \text { or } 8 > x \implies 8 \ge x \implies x \le 8.[/MATH]
But this is only half the problem.

Now work through [MATH]x - 13 \ge 5.[/MATH]
 
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