Question

Find the solution of the inequality.
|x-5|greater than or equal to 3

Answer 1

So geometrically, it's all numbers x which are a distance of 3 or more from 5. What numbers are those?

Answer 2

no its this sign > with a line under it so thats why i put greater than or equal to

Answer 3

Yes, so at a distance 3 or more from 5. For example 8 and all numbers greater than 8.

Hence one part of the solution is the real numbers \( x \) such that \( x \geq 8 \). Now, what's the other part?

Answer 4

I have no clue I don't understand this whole question at all

Answer 5

Let's step back. What is the definition of |x|?

Answer 6

?

Answer 7

What does
|x|
mean?

Answer 8

um again i don't know

Answer 9

Ok. |x| is called the absolute value of x. It measures how far the number x is from zero on the real number line.

How far is 1 from 0? 1. Hence |1| = 1.

How far is -1 from 0? 1. Hence |-1| = 1.

How far is 0 from 0? 0. Hence |0| = 0.

Now what is
|2| and |-2| ?

Answer 10

0?

Answer 11

No, 2 is a distance of 2 units away from zero. Hence
|2| = 2.

What then is
|-2| = ?

Answer 12

2

Answer 13

Yes. Now what is |-5|?

Answer 14

5

Answer 15

Yes. And what is |2-1|? This is the distance between 2 and 1

Answer 16

1?

Answer 17

Yes. And what is the distance between 1 and -1?
|1-(-1)| = ... ?

Answer 18

0?

Answer 19

Is 1 = -1? I don't think so. Try again.
|1-(-1)| = ... ?

Answer 20

1

Answer 21

Draw a number lilne. Locate -1 and locate +1. How far apart are they?

Answer 22

well its -1 then 0 then 1

Answer 23

Yes, so they are a distance of 2 apart.

I.e.,
|1-(-1)| = |1+1| = 2.

Answer 24

ohhh ok

Answer 25

Now, what number or numbers x are a distance 2 apart from 1?
i.e.,
|x-1| = 2.

Answer 26

1?

Answer 27

How far apart is 1 from 1? A distance of zero. Hence if x=1, then
|x-1| = 0.

We want numbers x such that
|x-1| = 2.

What's one number x that is a distance 2 away from 1?

Answer 28

look at a number line.

Answer 29

ughhh i just don't understand this

Answer 30

Draw a number line. Mark 0 and 1 and the numbers -5 to 5 on it.

Now look at 1. What are the numbers a distance 2 to the left of 1 and 2 to the right of 1?

Answer 31

x is less than or equal to 8 or x is greater than or equal to 2 this is what I put on my test

Answer 32

We're coming back to your problem next. But answer this one first.

If x is a real number with the property that
|x-1| = 2
i.e., x is a distance of 2 away from +1,
what values can x have?

Answer 33

-1 and 3?

Answer 34

3 is one such number. 3 is a distance 2 away from +1.

This makes sense as |3-1| = |2| = 2.

Now, what's the other number x such that
|x-1| = 2
?

Answer 35

Yes -1 and 3.

Answer 36

returning now to your problem, we are told that
\[ |x-5| \geq 3 \]
Let's first solve
\[ | x - 5| = 3 \]
What two numbers satisfy this last equation?

Answer 37

i.e., x is a distance 3 from +5.

Answer 38

so that would be 2?

Answer 39

That's one number. What's the other?

Answer 40

-2?

Answer 41

No. the difference between -2 and 5 is 7. I.e.,
|(-2)-5| = |-7| = 7.

Try again.
|x-5| = 3

Answer 42

is it 8

Answer 43

Yes of course. 8 and 2 are both a distance 3 from 5.

i.e., if
|x-5| = 3, then
x = 3 or 8.

Answer 44

Now you want real numbers x such that
\[ |x-5| \geq 3 \]
So the distance is equal to or \( \it greater \) than 3. What's a number more than 3 away from 5?

Answer 45

4

Answer 46

9 is one such number. So is 10 and 11 and 20 and 200 and 2000 and 50000 and 100000.

So is 8.5 and 8.1 and 8.00001 and 8.0000000001.

Answer 47

Hence *every* number greater than 8 is more than a distance of 3 away from 5.

Answer 48

so 9?

Answer 49

Look at the number line. *EVERY* number greater than 8 is more than a distance 3 away from 5. Hence EVERY number greater than or equal to 8 satisfies
\[ |x-5| \geq 3 \]
make sense?

Answer 50

kinda

Answer 51

im working on it might be getting there but slowly lol

Answer 52

Now, look on the other side of 5. What are the numbers x which are a distance of 3 or more from 5?

Answer 53

We know that 2 is one of them. What other numbers?

Answer 54

8?

Answer 55

is that a question or statement?

Answer 56

Draw a number line. Mark on it 0 and 5 and 10 and -5 and -10.

Answer 57

Tell me when you've done that.

Answer 58

In fact, mark on it all the numbers from -10 to 10.

Answer 59

k hold on

Answer 60

ok i got it

Answer 61

ok. Now look at 5. What numbers are 3 or more away from 5. Well, on the right hand side, clearly 8 is at least a distance 3 away from 5.

And in fact, so must every number greater than 8. So far so good?

Answer 62

yes

Answer 63

Now, to the left of 5, what number is *exactly* a distance of 3 away from 5?

Answer 64

2?

Answer 65

Yes 2. Now staying to the left of 5, what numbers are a distance of more than 3 away from 5?

Answer 66

uhh.. going 3 distance away from would be 2 or if your asking to go 3 more away from five would be -1

Answer 67

I am asking for numbers to the left of 5 on the number line, in other words for numbers less than 5, what is an example of a number which is a distance of more than 3 away from 5?

Answer 68

For example, 1 is at a distance of 4 from 5. Hence x=1 satisfies
\[ |x-5| \geq 3 \]
Yes?

Answer 69

so it would be 4?

Answer 70

because in fact
|1-5| = |-4| = 4.

That is, the distance between 5 and 1 is 4.

Now what are all the numbers less than 5 which are at a distance of more than 3?
Look at the number line. You know that 2 is *exactly* 3 from 5. What numbers are more than a distance of 3 from 5?

Answer 71

3 & 4?

Answer 72

omg im so confused

Answer 73

the distance between 3 and 5 is 2:
|3-5| = |-2| = 2

The distance between 4 and 5 is 1:
|4-5| = |-1| = 1

So no, that cannot be right.

Answer 74

Again, what numbers are *exactly* a distance of 3 away from 5? 8 and 2.

Agreed?

Answer 75

yes, it is and would it be 1?

Answer 76

So if the question was: for what values of x is
|x-5| = 3
the answers would be 2 and 8, and we'd be finished.

But that's not the problem.

Answer 77

The problem is: For what values of x is
\[ |x-5| \geq 3 \]
That is, what numbers x are at least a distance of 3 away from 5.

Answer 78

sorry im taking all of your time im just dumb i guess..

Answer 79

Now, clearly 8 is at least 3 away from 5 and hence every number greater than 8 is more than a distance of 3 away from 5.

So far so good?

Answer 80

yea

Answer 81

Now what other number is *exactly* a distance of 3 away from 5?

Answer 82

2?

Answer 83

i dont get how there is another number exactly a distance of 3 away from 5

Answer 84

Yes 2. Now what other numbers to the left of 5 are more than 3 units away from 5?

Answer 85

Look at the number line. You have to go 3 units from 5 to 8. And 3 units from 5 to 2.

Absolute value measures distance, not direction, not plus/minus. All we care about here is measuring distance.

Answer 86

I don't know what that means. But +1 is more than 3 units away from 5. So is 0, so is -1.

So is 1.5, so is 1.9.

In fact, every number less than 2 is more than 3 units distance from 5.

Make sense?

Answer 87

thats more less what i just said every number away from 2 is more than 3 units away from 5

Answer 88

less than 2, right.

So let's pull this altogether: What are all the real numbers x such that
\[ |x-5| \geq 3 \]?

Answer 89

2 and 8

Answer 90

No, that's the solution to the question:
What are the real numbers x such that
\[ |x-5| = 3 \]
?

Answer 91

thats what the question is about finding the solution

Answer 92

Your original question was:
"Find the solution of the inequality.
|x-5|greater than or equal to 3"

So you need the inequalities as well.

In other words, you don't want just the x such that
\[ |x-5| = 3 \]
We also want the x such that
\[ |x-5| > 3 \]

Answer 93

In other words, we don't just want the x which are exactly a distance 3 from 5, we also want all the x which are a distance greater than 3 from 5.

Answer 94

ok

Answer 95

would it be |2-5|=3

Answer 96

x=2 is one solution. But it's not all the solutions.

Answer 97

Look at your number line.

Answer 98

We want all of the numbers x which are a distance of 3 or more from 5.