The sum of 9 times a number and the reciprocal of the number is positive. Find the numbers which satisfy this condition
To find the numbers that satisfy the given condition, we need to set up an equation and solve for the number. Let's first represent the unknown number as "x".
According to the given condition, the sum of 9 times the number and the reciprocal of the number is positive. Mathematically, this can be written as:
9x + 1/x > 0
To solve this equation, let's first get rid of the fraction by multiplying both sides by "x":
9x^2 + 1 > 0
Now we have a quadratic inequality. To solve it, we need to find the values of "x" for which the inequality holds true.
Let's analyze the quadratic equation:
9x^2 + 1 > 0
Since the coefficient of x^2 is positive (9 > 0), the parabola opens upwards. Therefore, the graph of this quadratic equation will be a U-shaped curve that lies above the x-axis.
To find the solutions, we need to determine where this curve is above the x-axis. We can use various techniques, such as factoring, completing the square, or using the quadratic formula. However, in this case, the quadratic equation does not factor easily. Therefore, we'll use another approach.
Let's solve the equation by using the discriminant. The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
In our case, the quadratic equation is 9x^2 + 1 = 0. So, a = 9, b = 0, and c = 1.
The discriminant for this equation is D = 0^2 - 4(9)(1) = -36.
Since the discriminant is negative, D < 0, it means that the quadratic equation has no real solutions. This implies that there are no real numbers that satisfy 9x + 1/x > 0 for the given condition.
Therefore, there are no numbers that satisfy the given condition.
Let's represent the number as x.
According to the given condition, the sum of 9 times the number (9x) and the reciprocal of the number (1/x) is positive.
Mathematically, we can write this as:
9x + 1/x > 0
To solve this inequality, we can examine the different cases:
Case 1: x > 0
If x is positive, then 1/x is also positive.
In this case, 9x + 1/x will be positive for any positive value of x.
So, any positive value of x will satisfy this condition.
Case 2: x < 0
If x is negative, then 1/x is negative.
In this case, 9x + 1/x will be negative for any negative value of x.
So, no negative value of x will satisfy this condition.
Case 3: x = 0
If x is 0, then the expression 1/x is not defined.
Thus, x = 0 does not satisfy this condition.
Therefore, the numbers that satisfy this condition are any positive value of x.