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 what this’s true
The number ---- x
the reciprocal of the number --- 1/x
"The reciprocal of a number decreased by the reciprocal of twice the number"
---- 1/x - 1/(2x)
1/x - 1/(2x) ≥ 2
2/2x - 1/2x ≥ 2
1/x ≥ 2
1 ≥ 2x
2x ≤ 1
x ≤ 1/2
test:
let x = 1/4
reciprocal --- 4
twice the number --- 1/2
reciprocal of twice the number --- 2
4 - 2 ≥ 2, yes!
let x = 5
reciprocal --- 1/5
twice the number --- 10
reciprocal of twice the number --- 1/10
1/5 - 1/10 ≥ 2, no!, and that is what it should be
To solve this problem, let's break it down step by step:
Let's assume the number we are looking for is "x".
According to the problem, the reciprocal of "x" decreased by the reciprocal of twice the number (2x) is greater than or equal to 2.
So, we can start by writing this inequality:
1/x - 1/(2x) ≥ 2
Now, let's find a common denominator for the fractions on the left side of the inequality. The common denominator is 2x:
(2 - 1) / (2x) ≥ 2
Simplifying the numerator:
1 / (2x) ≥ 2
Now, cross multiply to get rid of the denominator:
1 ≥ 4x
Rearrange the inequality:
4x ≤ 1
Divide both sides by 4:
x ≤ 1/4
Therefore, the solution to the inequality is any number less than or equal to 1/4.