Inequation with just 3 solutions

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  • Thread starter Vali
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In summary, the function f(x)=log_3(x) (with a base of 3) has an inequation of f(n/x) >= 1. To have just 3 solutions in N*, n must be greater than or equal to 3x. The number of n values that satisfy this condition is 3.
  • #1
Vali
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I have the following function:
f: (0,infinity) -> R
f(x)=log_3(x) (the base is 3)
I need to find the number of "n" values ( n is a natural number except 0 N*) such that this inequation: f(n/x) >= 1 to have just 3 solutions in N*.
A. infinity
B. 6
C. 9
D. 26
E. 3 (correct answer)
 
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  • #2
Maybe I'm reading this wrong, but it looks to me like you need to count the number of natural numbers greater than \(3x\), which for any choice of \(x\) allowed, would be countably infinite.
 
  • #3
Yes, I got the same result n>=3x
Maybe I wrote the sentence is a wrong way because I translated it from romanian.
I posted o picture below.
Exercise number 45.
The number of n values ( n natural ) for which the inequation f(n/x) >= 1 has exactly 3 solutions in N* is: ...
 

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  • #4
Vali said:
I have the following function:
f: (0,infinity) -> R
f(x)=log_3(x) (the base is 3)
I need to find the number of "n" values ( n is a natural number except 0 N*) such that this inequation: f(n/x) >= 1 to have just 3 solutions in N*.
A. infinity
B. 6
C. 9
D. 26
E. 3 (correct answer)
This is a weird one. Here's my guess.
\(\displaystyle log_3 \left ( \dfrac{n}{x} \right ) \geq 1\)

Since the log function is continuous we can take the exponent of base 3 on both sides:
\(\displaystyle 3^{ log_3 (n/x) } \geq 3^1\)

\(\displaystyle \dfrac{n}{x} \geq 3\)

\(\displaystyle n \geq 3x\)

So to have only 3 solutions, n = 3, gives us a domain for x as {1, 2, 3}.

-Dan
 
  • #5
Thank you very much for the help :)
 

Related to Inequation with just 3 solutions

1. What is an inequation with just 3 solutions?

An inequation with just 3 solutions is a mathematical statement that compares two quantities using symbols such as <, >, ≤, or ≥, and has exactly 3 possible solutions that satisfy the inequality. These solutions are the values of the variable that make the inequality true.

2. How do you solve an inequation with just 3 solutions?

To solve an inequation with just 3 solutions, you first need to isolate the variable on one side of the inequality. Then, you can use the properties of inequalities to manipulate the inequality and find the three possible solutions. It is important to note that the direction of the inequality symbol may change depending on the operations used to solve it.

3. Can an inequation with just 3 solutions have no solution?

Yes, it is possible for an inequation with just 3 solutions to have no solution. This occurs when the inequality is not satisfied by any value of the variable. For example, the inequality 2x > 10 has no solution because there is no value of x that can make it true.

4. What is the difference between an equation and an inequation with just 3 solutions?

An equation is a mathematical statement that shows that two expressions are equal, while an inequation with just 3 solutions shows that one expression is greater than or less than another. Additionally, an equation can have an infinite number of solutions, while an inequation with just 3 solutions has exactly 3 solutions.

5. How can inequations with just 3 solutions be applied in real life?

Inequations with just 3 solutions can be used to represent and solve various real-life problems. For example, they can be used to determine the number of items that need to be sold in order to make a profit, or to find the range of values for a variable in a scientific experiment. They can also be used in economics, engineering, and other fields to make decisions based on inequalities.

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