SOLUTION: How many integers satisfy |x+1|<4

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Question 1207257: How many integers satisfy |x+1|<4
Found 5 solutions by math_tutor2020, greenestamps, ikleyn, Edwin McCravy, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 7

Work Shown

|x+1| < 4
-4 < x+1 < 4
-4-1 < x+1-1 < 4-1
-5 < x < 3

In the 2nd step I used the rule that |x| < k leads to -k < x < k for some positive real number k.

-5 < x < 3 means that x is between -5 and 3 excluding both endpoints.
If x was an integer then x can be selected from the set {-4, -3, -2, -1, 0, 1, 2}
There are 7 integers listed in that set.

Example: Let's check x = -2
|x+1| < 4
|-2+1| < 4
|-1| < 4
1 < 4
That works out.
I'll let the student check the other integer solutions.
Now let's try a non-solution such as x = 5
|x+1| < 4
|5+1| < 4
|6| < 4
6 < 4
Which is false so x = 5 is confirmed to be not part of the solution set.

Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


The response from the other tutor shows a standard formal algebraic solution.

Here is another way of solving absolute value equations that is often easier than the formal algebraic solution.

The expression



can be interpreted as meaning that the difference between x and a is less than b.

In this problem....

-->

This says that the difference between x and -1 is less than 4, where x is an integer. In other words, if we look at a number line, x can be any integer whose distance from -1 is less than 4.

3 units to the right of -1 is -1+3=2; 3 units to the left of -1 is -1-3=-4. So x can be any integer from -4 to +2 inclusive; that is 7 integers.

Or, the previous paragraph more simply: the integers that satisfy the inequality are -1, plus 3 integers either side of -1, for a total of 1+3+3 = 7.
Answer by ikleyn(52855)   (Show Source): You can put this solution on YOUR website!
.
How many integers satisfy |x+1| < 4 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Here is even more short reasoning: 

    The number of integer solutions  of  |x+1| < 4 is the same 
    as the number of integer solutions of   |y| < 4,
    which is obviously 3 + 3 + 1 = 7.

Solved.

----------------

Notice that in this problem,  they do not ask you to find the set of solutions.
They only ask to determine the number of solutions.

Therefore,  in this application,  x+1  is a dumb variable and can be replaced by any other variable.

This variable,  (x+1),  is intently presented in this form to create false associations,
to distract attention and to motivate a reader to make unnecessary calculations.

Therefore,  it is  MUCH  BETTER  to replace it by something more neutral, which does not
distract attention, does not make false associations and does not motivate to make unnecessary calculations.



Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

|x + 1| < 4

|x + 1| = 0, 1, 2, 3

 x + 1 = -3,-2,-1, 0, 1, 2, 3

So exactly 7 integers will satisfy the given inequality.
They are:
     
     x = -4,-3,-2,-1, 0, 1, 2 

Edwin
Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!
How many integers satisfy |x+1|<4

|x + 1| < 4 denotes: x + 1 < 4  or  x + 1 > - 4
When combined, we get: - 4 <   x + 1   < 4
                   - 4 - 1 < x + 1 - 1 < 4 - 1 ----- Subtracting 1
                           - 5 < x < 3
The number of INTEGERS from - 5 to 3, INCLUSIVE, is: 3 - - 5 + 1 = 3 + 5 + 1 = 9. However,
these 9 INTEGERS include the 2 endpoints, - 5 and 3, which should be EXCLUDED. Therefore, 
number of INTEGERS between - 5 and 3, that satisfies |x + 1| < 4 is: 9 - 2 = 7.
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