Solving Inequality Problem: (|a|+|b|≥2(|c|+|d|))

  • Thread starter drago
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In summary, the conversation discusses a problem involving an inequality and variables a, b, c, and d. The question is whether the inequality can lead to a conclusion about a specific equation. The participants share their thoughts and one suggests a weaker result that can be proven easily by making additional assumptions. However, this result is not as strong as the desired conclusion.
  • #1
drago
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Hi,
I would appreciate some ideas on the following problem:

We are given the inequality:
(1)
|a| + |b| >= 2(|c| + |d|)

Can we conclude that:
(2)
(c + a)^2 + (d + b)^2 >= c^2 + d^2 ?

a, b, c, d are real.

I can prove (2) if (1) is in the form: |a| + |b| >= 4max(|c|,|d|), but the above is stronger.

Thank you.
drago
 
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  • #2
i feel its true
its seems so easy but at 12.30 midnight i cannot think of anything! :frown:

-- AI
 
  • #3
i feel its true
its seems so easy but at 12.30 midnight i cannot think of anything!

Yep I looked at it for a few minutes late last night and thought it looked easy but couldn't come up with much. I got another weaker result pretty easy but then decided to leave it for someone else.

BTW, I got the weaker result by assumming (without loss of generality) that |b| >= |a|. I then found that if I made the additional imposition that |c|>=|d| then I was able to prove the desired result pretty easily . This however definitely does give a loss of generality and hence a weaker result.
 

1. How do you solve an inequality problem involving absolute values?

To solve an inequality problem involving absolute values, you first need to isolate the absolute value expression on one side of the inequality. Then, you can split the inequality into two separate cases based on the positive and negative values of the absolute value expression. Finally, you can solve each case separately and combine the solutions to find the overall solution for the original inequality.

2. What is the purpose of splitting the inequality into two cases?

Splitting the inequality into two cases allows us to consider both the positive and negative values of the absolute value expression. This is necessary because the absolute value of a number can be equal to either the number itself or its negative value.

3. Can we use the same method to solve all inequality problems involving absolute values?

No, the method for solving inequality problems involving absolute values may vary depending on the specific problem. In some cases, we may need to use different strategies such as graphing or substitution to find the solution.

4. How can we check if our solution is correct for an absolute value inequality problem?

To check if our solution is correct, we can substitute the values into the original inequality and see if it holds true. Alternatively, we can graph the inequality and see if our solution falls within the shaded region on the graph.

5. Are there any special rules to keep in mind when solving absolute value inequality problems?

Yes, there are a few rules to keep in mind when solving absolute value inequality problems. For example, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be flipped. Also, when taking the square root of both sides, we need to consider both the positive and negative values of the result.

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