How would you determine if the following problem is =, >, or <
x____________x^2, where x is a real num-
ber and 0 < x < 1.
To determine if the expression x is equal to, greater than, or less than the expression x^2, where x is a real number between 0 and 1, we can evaluate both expressions for different values of x within the given range.
Let's start by substituting x = 0.5 into the expressions:
For x, we have: x = 0.5
For x^2, we have: x^2 = (0.5)^2 = 0.25
Since 0.5 is greater than 0.25, we can conclude that x > x^2 when x = 0.5.
Now, let's substitute x = 0.2:
For x, we have: x = 0.2
For x^2, we have: x^2 = (0.2)^2 = 0.04
Since 0.2 is less than 0.04, we can conclude that x < x^2 when x = 0.2.
By analyzing these examples, we can see that when x is between 0 and 1, the value of x^2 is always greater than x. Therefore, we can state that x < x^2 for all real numbers x, where 0 < x < 1.