https://file.aiquickdraw.com/mathbot/file/17259146876669nkx3gbf.jpg

The image shows a question in German that provides three mathematical statements, and you are asked to select which ones are correct. The question involves real numbers \(a\), \(b\), and \(c\). Let's break down each statement:

1. **Statement 1**: \( a < b \Rightarrow -a < -b \)
   - This is **incorrect**. If \(a < b\), negating both sides reverses the inequality, meaning \( -a > -b \), not \( -a < -b \).

2. **Statement 2**: The inequality \(a - 3 < 1\) has the solution set \(L = (4, \infty)\).
   - Let's solve the inequality:
     \[
     a - 3 < 1
     \]
     Add 3 to both sides:
     \[
     a < 4
     \]
     The correct solution set should be \( (-\infty, 4) \), not \( (4, \infty) \). So this statement is **incorrect** as well.

3. **Statement 3**: \( a < b \Rightarrow a + c < b + c \)
   - This is **correct**. Adding the same number \(c\) to both sides of the inequality does not change the inequality's direction. Therefore, \( a + c < b + c \) holds true.

### Conclusion:
- The only correct statement is the third one: \( a < b \Rightarrow a + c < b + c \).

Do you want further details on any part of the explanation?

Here are 5 related questions:
1. What happens when we multiply or divide both sides of an inequality by a negative number?
2. How does shifting an inequality by adding or subtracting the same value affect it?
3. Can we apply similar rules for inequalities in non-real number systems (e.g., complex numbers)?
4. How can you verify if an inequality solution is correct by graphing it?
5. What are the differences between strict inequalities (e.g., \(<\)) and non-strict inequalities (e.g., \(\leq\))?

**Tip**: Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Given a, b, c ∈ ℝ. Which statements are correct? Select one or more answers:
1. a < b ⇒ -a < -b
2. The inequality a - 3 < 1 has the solution set L = (4, ∞).
3. a < b ⇒ a + c < b + c

Learn how to evaluate the correctness of statements involving inequalities in real numbers, including properties such as negation and addition. This problem is suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation