How to Select the Correct X-Value for Testing Inequalities?

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Discussion Overview

The discussion revolves around solving the inequality 2x - 7 < 11 and the appropriate methods for verifying the solution. Participants explore the implications of selecting specific x-values for testing the validity of the inequality, focusing on the correctness of the derived solution x < 9.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the solution to the inequality as x < 9 and checks it by substituting x = 0, concluding it is correct.
  • Another participant agrees with the initial solution and confirms that any number less than 9 satisfies the inequality.
  • However, a later reply challenges the method of checking by stating that using a single value (like x = 0) does not prove that all numbers less than 9 satisfy the inequality.
  • This participant emphasizes the need for a more rigorous approach to selecting x-values for testing the inequality.

Areas of Agreement / Disagreement

While some participants agree on the solution x < 9, there is disagreement regarding the method of verification. The discussion highlights differing views on how to appropriately select x-values for testing the inequality.

Contextual Notes

The discussion does not resolve the question of how to select the correct x-value for testing, leaving the method of verification open to interpretation and further exploration.

mathdad
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Solve the inequality.

2x - 7 < 11

2x < 11 + 7

2x < 18

x < 18/2

x < 9

Correct?
 
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It's easy to check. If x is any number less than 9 then 2x is less than 18 so that 2x- 7 is less than 11. Yes, that is correct.
 
Good to be correct.

2x - 7 < 11

The value of x must be less than 9 to make the original inequality a true statement.

Let x = 0

2(0) - 7 < 11

0 - 7 < 11

-7 < 11

This is true. So, x < 9 is correct.
 
RTCNTC said:
Good to be correct.

2x - 7 < 11

The value of x must be less than 9 to make the original inequality a true statement.

Let x = 0

2(0) - 7 < 11

0 - 7 < 11

-7 < 11

This is true. So, x < 9 is correct.
No, that is not an appropriate way to check. That shows that there exist a number, less than 9, that satisfies the equation. It does not show that every number less than 9 satisfies it.

For example, suppose you had arrived at the incorrect conclusion that the solution was x< 5. Taking x= 0, which is still less than 5, would arrive at the same result.
 
Not every number less than 9 can be used to show that the original inequality is true. How does one select the correct x-value for testing?
 

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