If |x–1|<4, so:

Bunnel,

First of all, thank you very much.

The source of this questions is a bank of questions that I received from a friend, whose his teacher gave to him.

I have huge doubts on inequalities topics.

As you explained above any value X>-4 can satisfy this.

What is the best approach to deal with this kind of questions? I solved -3<x<5, but I could not figure out, X>-4 because -3>-4 and do not satisfy the inequality.

Can you please help?

I have the same doubt on this question, below...

https://gmatclub.com/forum/if-z-2-4z-5- ... l#p1949237

It's the other way around. We know that \(-3<x<5\). Any x from that range satisfy x > -4.


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\(|x–1|<4\);

\(-4<x-1<4\);

\(-3<x<5\).

Any x from that range will be more than -4.

Answer: D.

P.S. What is the source of this question?[/quote]



Hi, Bunuel
Just wanted to ask that since in this question it is not mentioned that x is an integer, by the range x>-4 it will include values greater than -3 also (say -3.8) but our range is from -3 to 5 so isnt this wrong????

I tried to explain this in the post above.

We know that \(-3<x<5\). Any x from that range satisfy x > -4.


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We can solve the inequality when (x - 1) is positive and when it is negative.

When (x - 1) is positive:

x - 1 < 4

x < 5

When (x - 1) is negative:

-x + 1 < 4

-x < 3

x > -3

-3 < x < 5

Answer: D

It's a tricky question, I think I've encountered a sort of this one in OG2017.
Key to solve it is to find the range from the answer choices that covers all possible values of x,here it is answer choice D.

I can't understand why x>-4 is the correct answer.

Since we have -3<x<5 , -3 is out of the range and in x> -4 -3 is included in the number line.

Besides that, X>-4 we can have x=10, as an exemple, wich is not true since x<5.

Can someone help me with this points?

tks

Hi Bunuel, thanks for the explanation. I tend to approach this questions in the wrong way, trying to find an answer choice that matches with the range found through the inequalities. For instance, in this case I was looking for -3<x<5 (or something more restrictive, such as -2 < x < -5), which led me to pick answer E.

Given your explanation, I understand that I should look for that condition that holds for every x in the range...but I still have a huge doubt.
You say it is (D) x > -4 because any X from that range will be more than -4. Following your logic, wouldn't also B) x ≤ 4 be true? Our range is -3<x<5, which means that every x will be less or equal to 4.

Is there something I am not getting right?

Your help is super appreciated

For \(-3<x<5\), (B) x ≤ 4 won't always be true. For example, x could be 4.5 (4.5 IS between -3 and 5) and in this case B is not true.

|x–1|<4


−3<x<5

Any x from that range will be more than -4.

Answer: D.

Check other similar questions from Trickiest Inequality Questions Type: Confusing Ranges (part of our Special Questions Directory).

Wouldn't this question be better off as a "which of the following must be true" question? Does the actual test do stuff like this?

On solving the inequality, we get x<5 and x>-3.
Among the options, we have x>-4, which is already satisfied given that x>-3 and since -4<-3, we can conclude that x>-4.
Hence, Option D is the right choice.

Although this is a relatively simple question on absolute values (modulus), the way in which the options have been framed can put you off.

The reason for this is that, you go in to the solution, expecting to see a range that you think you’ll get when you solve the given inequality. But, in reality, none of the options represent the exact range that you will get when you solve the inequality.
Therefore, it becomes important to understand that , here, we are trying to find out a range of values, which in turn fully covers the range which satisfies the given inequality.

If |x-a| < y, the range of x that satisfies the inequality will be a-y < x < a+y. When we apply this to the given inequality, |x-1| < 4, we can say that -3<x<5. Let us represent this on a number line, as shown below:



Let us evaluate the options now.

Option 1 says x>3. This means that x can be any value from 3 to infinity. Clearly, only a portion of this range satisfies our inequality i.e. 3<x<5.
So, quite obviously, you cannot say that all values which satisfy |x-1|<4 fall in the range of x>3.



A similar thing works against option B.

With option C, although some numbers are within the range, some numbers aren’t. Like for example, x = 4.5 satisfies the given inequality but is not within -4<x<4.

When it comes to option D, we can say that all the values that satisfy our inequality are greater than -4. This means, when I pick any value from the range satisfying the inequality, it will always qualify as true when compared with option D.

The same cannot be said about the other options, as we have demonstrated.
So, the correct answer is D.

The confusion that this question has created could have been nullified if the question was framed as a ‘Must be’ question. For example “ If |x-1|<4, which of the following must be true for the values of x that satisfy the inequality?” . Essentially, this is what the question is testing you on.

Hope this helps!

I am in a huge confusion. The range is -3<x<5 and I eliminated all the choices based on this range including D. Because x>-4 does not give an upper limit and it can be 10 or smth more. What I miss here?

Hi

A simpler and similar question would be

If x>2, then
‘x is positive’ is correct since x has to be positive. x>0 may contain extra values, those between 0 to 2, but it also encompasses the entire possible range.
Similar is the case here. We are not looking at exact range here, but at what MUST be true.

How? I dont get it.

When we write -3<x how can this be x>-4 🤷‍♂️. Isn't it mean x can be -3?

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