1. ## Inequalities

If 0<a ≤b ≤ c then b<a+c and a<b+c follow automatically.

I do not understand why b<a+c and a<b+c will follow automatically. Can you please explain? Thank you in advance!

2. Originally Posted by Anna55
If 0<a ≤b ≤ c then b<a+c and a<b+c follow automatically. I do not understand why b<a+c and a<b+c will follow automatically. Can you please explain? Thank you in advance!
Given that $\displaystyle 0<a\le b$ do you see why $\displaystyle b\le b+a~?$

3. Yes I understand your example. If I plug in values into the original inequalities, I see that it is correct, however I would not be able to write that the two inequalities would follow automatically since I do not understand that fully.

4. Originally Posted by Anna55
Yes I understand your example. If I plug in values into the original inequalities, I see that it is correct, however I would not be able to write that the two inequalities would follow automatically since I do not understand that fully.
Would you agree that if $\displaystyle 0<a\text{ then }x<a+x\text{ for any real number }x~?$

If you do then let $\displaystyle x=b~\&~x=c.$

5. I almost understand your example. This is my attempt to use your method on the original problem.
0<a≤ b≤c
I replace a and b with x.
0< a≤ x≤ x
a-x ≤ x-x
0<a-x<0<x
<a-x≤x

Where is my error or how can you continue with solving the inequality?

6. The original problem says nothing about subtraction. Why are you subtracting x?
Also, you cannot use "x" to represent both a and b which may be different numbers.

Do you understand that adding a positive number to any expression makes it larger? That is, if a> 0, b< b+ a for any b.

7. Originally Posted by Anna55
If 0<a ≤b ≤ c then b<a+c and a<b+c follow automatically.
I do not understand why b<a+c and a<b+c will follow automatically. Can you please explain? Thank you in advance!
$\displaystyle a>0$

$\displaystyle b\le\ c\Rightarrow\ b<\;\; any\; value\; greater\; than\; c$

$\displaystyle c+a>c$

Try the 2nd one.

8. Originally Posted by Anna55
If 0<a ≤b ≤ c then b<a+c and a<b+c follow automatically.

I do not understand why b<a+c and a<b+c will follow automatically. Can you please explain? Thank you in advance!

Perhaps you'll understand these inequalities better if you put a, b and c on x-axis.