The reciprocal of a number decreased by the reciprocal of twice the number is greater than or equal to 2. Find the number(s) for which this true.
1/x - 1/(2x) >= 2
2/(2x) - 1/(2x) >= 2
2-1 >= 2*2x
1 >= 4x
x <= 1/4
To solve this problem, let's assume the given number as "x".
The reciprocal of a number is obtained by taking the reciprocal of the number itself, which is 1 divided by the number. So, the reciprocal of "x" is 1/x.
According to the given information, "the reciprocal of a number decreased by the reciprocal of twice the number is greater than or equal to 2". Mathematically, this can be represented as:
1/x - 1/(2x) ≥ 2
To solve this inequality, we need to combine the fractions on the left side and then isolate the variable "x".
To combine the fractions, we need to find a common denominator. The common denominator in this case is 2x.
1/x - 1/(2x) can be rewritten as (2 - 1) / (2x), which simplifies to 1/(2x).
So now we have:
1/(2x) ≥ 2
To get rid of the fraction, we can multiply both sides of the inequality by 2x:
2x * 1/(2x) ≥ 2 * 2x
This simplifies to:
1 ≥ 4x
Next, divide both sides of the inequality by 4:
1/4 ≥ x
So, the solution to the inequality is x ≤ 1/4. This means that any number less than or equal to 1/4 will satisfy the given inequality.
Therefore, the numbers for which the inequality is true are x ≤ 1/4.