Inequality

[Casio]

Hi everyone

I have an inequality

2x - 4 < 1

I had to double check it to ensure I wrote it down correctly.

2x < 1 + 4

x < 2.5

2(2.5) - 4 < 1

1 < 1

Is this me or am I missing something?

2x - 4 < 1 reads to me as 2x - 4 should be less than < 1 and not equal to it?

 

[Fantini]

In order to check it you should try numbers less than $5/2$, not equal to. Once you plugged it in the original equation it was good that it wasn't a solution, or else something would have went horribly wrong. Try $x=2$.

 

[Casio]

Yes I see what you mean when putting 2 into the inequality, but I am making that figure up knowing it will be less than 1?

My misunderstanding seems to be that finding the value of 'x' in this example does not prove the inequality correct?

I must be missing something here as x = 2.5 but for some reason in this example 2x - 4 < 1 mathematically does not work?

2(2.5) - 4 < 1

Is it not a typo error?

should it not be;

2(2.5) - 4 < 1

 

[Fantini]

The values of $x$ you have found are the ones less than two and half, not equal to. Why should it be $2x+4 \leq 1$? You don't need equality. Geometrically, you have the points belonging to the line $y=2x+4$ and below the line $y=1$, but you discount the intersection, which happens at the point $x= 5/2$.

Also, note that $5/2$ is not less than itself, thus it cannot be a solution! If it doesn't belong to the solution set, it cannot satisfy the given inequality.

 

[MarkFL]

You have found that x must be less than 2.5, so as stated above, if you let x = 2.5, then your inequality will not be true.

Let x = 2.5 - y where y may be as small or large as we desire, as long as 0 < y.

Now, substituting this into the original inequality, we find:

2(2.5 - y) - 4 < 1

5 - 2y - 4 < 1

1 - 2y < 1

0 < 2y

0 < y

 

[Casio]

OK I think I have got it now. I find a value for 'x' which I did at ⁵/_2, which is in decimal form 2.5.

This value is definately in the inequality, so is a strick value. The misunderstanding I think I had was in understanding that ALL values up to 2.5 can be considered, so if I said;

x = - 2, which is < 2.5, I could write;


2(- 2) - 4 < 1


- 4 - 4 < 1


I understand it know, thanks everyone.