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My question is:

Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero.

What I have tried so far: If $a>0$ then: $x < a+4$ and $x>4-a$, if $a=0$ then there is no solution.

My doubt is: Should I consider the case $a<0$ as again $|x-4| which is not possible as absolute value cannot be negative.

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    Note that if $a \le 0$, there is **no** $x$ such that $|x-4|\lt a$, because $|w|$ is always $\ge 0$. That will com out if you use the general machinery described by robjohn, but you might as well handle it separately.2012-06-02
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    Probably the hint should be: depending on whether $x-4$ is negative, positive, or zero.2012-06-03

1 Answers 1

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Hint: $|x-4| means that $x$ is closer than $a$ units to $4$.

Another Hint: $|x-4| means that $(x-4) and $-(x-4).

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    I really dont understand these modulus based questions. plz can you try to show it on a number line?2012-06-02
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    There is no modulus here; there is an [absolute value](http://en.wikipedia.org/wiki/Absolute_value). Try this: draw a [number line](http://en.wikipedia.org/wiki/Number_line) from $0$ to $10$, then color the part of the number line that is closer than $2$ units to $4$. Try the same thing for the part closer than $3$ units to $4$.2012-06-02
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    @ robjohn ya I did that2012-06-02
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    @ robjohn Plz can you tell me the solution as I am getting very confused.2012-06-02
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    @user1396721: using the second hint, can you plot $(x-4) and $-(x-4)? The answer is the intersection of these sets.2012-06-02
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    @ robjohn thanks a lot2012-06-02
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    But thats when 'a' is positive right?2012-06-02
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    @user1396721: no, that holds for all $a$. Think of $|x|$ as $\max(x,-x)$.2012-06-02
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    @ robjohn:the only case which can be considered for 'a' is a>0 right? as if a=o then |x-4|<0 which is not possible and same is for 'a'<02012-06-03
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    "modulus" is another word for "absolute value", robjohn. Yes, the same word is used in connection with congruences, but there are only so many words to go around, some of them have to work overtime.2012-06-04
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    @Gerry: I see that in the link I provided, it says "In mathematics, the **absolute value** (or **modulus**)". I apologize for being so narrow-minded.2012-06-04