Help needed with inequality

[Catherine]

If you have the book, it's the second lesson of chapter 13, Inequalities, in Paul Foerster's Algebra 1 book.  

 

The example:

 

3x+11<3(x+7)   BUT it says less than or equal to, not less than.  I can't find that symbol on my computer.  

 

The solution is 11<21 (less than or equal to), or the set of all real numbers.

 

In the problems for this section, there is a similar problem:

 

4x<37+4x   Again the symbol in the book is less than or equal to.  The answer again in the set of all real numbers.  

 

What my son and I don't understand is-why is 11=21 and 0=37 in these problems, respectively?  I would assume the less than or equal to sign was a typo if it didn't show up in 2 different problems.  

 

Can anyone explain?  

 

Can anyone explain how I taught this twice before and did not catch it? (I'm embarrassed!)

[Cosmos]

They aren't equal. Any real number will make the inequality true by satisfying the "less than" part. If it JUST had an equal sign, as in

 

3x + 11 = 3 (x + 7)

 

then there would be no solutions. But since it's written with a "less than or equal" (which is usually type-written like this: <=)

 

3x + 11 <= 3 (x + 7)

 

then you only need the less than to be true OR the equals to be true. It's less than OR equal. Since the "less than" part is obviously true for all real numbers, the inequality as a whole is true for all real numbers.

[underthebridge]

I don't see where the problems state that 11=21 and 0=37. "≤" indicates less than OR equal which is the same as NOT GREATER THAN

 

In the first problem, the equation reduces to 11≤ 21, which is true. 

 

In the second problem, the book seems to be showing that if you add a constant to each side of an inequality equation the relationship remains the same

 

4x â‰¤ 37 + 4x

0 â‰¤ 37

 

 

[Catherine]

OR!!!!

 

Of course.  

 

I think I need another cup of tea.