Half of a number is at most 5 inits from 14
Half of a number n is at most 5 units from 14 means:
Difference between 14 and half times n is less than 5.
| 14 - n / 2 | ≤ 5
Multiply both sides by 2
| 28 - n | ≤ 10
Two cases are possible:
28 - n ≤ 10
OR
- ( 28 - n ) ≤ 10
1.
28 - n ≤ 10
28 ≤ 10 + n
28 - 10 ≤ n
18 ≤ n
2.
- ( 28 - n ) ≤ 10
- 28 + n ≤ 10
n ≤ 10 + 28
n ≤ 38
Solution:
18 ≤ n ≤ 38
|1/2n - 14| ≤ 5
Half of n is 1/2 * n or 1/2n, you need to subtract 14 from that and you put all of it in absolute value signs. Then, since it says is at most 5, you know it can be 5 or less which is shown by ≤ 5.
Oh, the number is a bit shy, isn't it? It likes to stay within a safe distance from 14. Well, if half of the number is at most 5 units away from 14, then we can set up an inequality:
|x/2 - 14| ≤ 5
This means that the absolute value of half of the number minus 14 is less than or equal to 5. So, we can solve this inequality to find the range of possible values for the number.
To solve this problem, we need to find a number where half of it is at most 5 units away from 14.
Let's call the number "n". The problem statement tells us that half of the number is at most 5 units away from 14. Mathematically, we can write this as:
|n/2 - 14| ≤ 5
To simplify the equation, we can look at two cases: when n/2 - 14 is positive and when it is negative.
Case 1: n/2 - 14 is positive
In this case, we can rewrite the equation as:
n/2 - 14 ≤ 5
Adding 14 to both sides, we get:
n/2 ≤ 19
Multiplying both sides by 2, we find:
n ≤ 38
So if n is positive, it must be 38 or less.
Case 2: n/2 - 14 is negative
In this case, we can rewrite the equation as:
-(n/2 - 14) ≤ 5
Expanding the negative sign, we have:
- n/2 + 14 ≤ 5
Subtracting 14 from both sides, we find:
- n/2 ≤ -9
Multiplying both sides by -2 (which reverses the inequality because we are multiplying by a negative number), we get:
n ≥ 18
So if n is negative, it must be 18 or greater.
Combining these results, we find that the number "n" can be any value from -∞ to -18 inclusive, or from 18 to +38 inclusive.