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

[mathdad]

Solve the inequality.

2x - 7 < 11

2x < 11 + 7

2x < 18

x < 18/2

x < 9

Correct?

 

[HallsofIvy]

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.

 

[mathdad]

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.

 

[HallsofIvy]

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.

 

[mathdad]

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?