# Inequalities help

#### Bogogogo

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

#### topsquark

##### Full Member

-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

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

View attachment 13559
The expression $$\displaystyle |x-13|$$ is read "the distance from $$\displaystyle x\text{ to }13$$".
So what is the inequality?

#### topsquark

##### Full Member
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

##### New member
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

##### Elite Member
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

##### Elite Member
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

##### New member
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

##### Elite Member
You are guessing.

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

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

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 $$\displaystyle x - 13 \ge 5.$$