Proofs involving real numbers

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Max.Maxim

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Let x and y be real numbers such that |x| ≤2 and |y| ≤4. Prove that 2x+ 3y−5|<28. Hello, I'm stuck in this problem and need help. Thank you!
 
Max.Maxim said:
Let x and y be real numbers such that |x| ≤2 and |y| ≤4. Prove that 2x+ 3y−5|<28. Hello, I'm stuck in this problem and need help. Thank you!
Click to expand...
What is the maximum value of:

x and

y and

2x + 3y ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.
 
Just build it up! It is not hard once you see the pattern

For example show |2x + 3y -11|<52
|x|<7 means -7 < x <7
|y|<9 means -9 < y <9
-11 means -11=-11=-11

Then:
-14 < 2x <14
-27 < 3y <27
-11 = -11 =-11
Adding gives us
-52 < 2x+3y-11< 30

So |2x+3y-11|< 52
 
Subhotosh Khan said:
What is the maximum value of:

x and

y and

2x + 3y ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.
Click to expand...
You realize that the OP meant to have (both) absolute value bars in Prove that |2x+ 3y−5|<28 which means that your response is incorrect.
 
Last edited:
Max.Maxim said:
Let x and y be real numbers such that |x| ≤2 and |y| ≤4. Prove that 2x+ 3y−5|<28. Hello, I'm stuck in this problem and need help.
Click to expand...
I am truly puzzled by this post. Look at this:
\(\begin{gathered} \left| x \right| \leqslant 2 \Rightarrow - 2 \leqslant x \leqslant 2 \Rightarrow - 4 \leqslant 2x \leqslant 4 \hfill \\
\left| y \right| \leqslant 4 \Rightarrow - 4 \leqslant y \leqslant 4 \Rightarrow - 12 \leqslant 3y \leqslant 12 \hfill \\
- 16 \leqslant 2x + 3y \leqslant 16 \Rightarrow - 21 \leqslant 2x + 3y - 5 \leqslant 11 \Rightarrow \left| {2x + 3y - 5} \right| \leqslant 21 \hfill \\
\end{gathered} \)
 
Jomo, this is what I did
Proof: let |x| ⩽ 2 and y ⩽ 4
So | 2x+3y-5| ⩽ |2x| + |3y| + |-5|
=2|x| + 3|y| + 5
=2(2) + 4(3)+5
=4 + 12 + 5
= 21
Therefore |2x+3y-5|<28
 
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