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

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To solve the inequality 2x - 7 < 11, the correct value of x must be less than 9. Testing with x = 0 confirms this, as it satisfies the inequality. However, simply checking one value is insufficient to prove that all numbers less than 9 work; it only demonstrates that at least one does. The discussion emphasizes the importance of selecting appropriate x-values for testing inequalities to ensure comprehensive validation. Ultimately, the conclusion is that x must be less than 9 for the original inequality to hold true.
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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?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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