Find the smallest integer x where the expression srt(x-2)/x^2+x-6 is defined. I factored the denominator (x+3)(x-2) and the smallest value is 2 so the whole expression would have a value of 0. But, that is not the right answer. Thank you!
To find the smallest integer x where the expression (sqrt(x-2))/(x^2+x-6) is defined, you need to consider the domain of the expression.
You correctly factored the denominator as (x+3)(x-2), which means the expression will be undefined when either x+3 or x-2 is equal to zero. So, we have two cases to consider:
Case 1: x+3 = 0
Solving for x in this case, we get x = -3.
Case 2: x-2 = 0
Solving for x in this case, we get x = 2.
Therefore, based on your calculations, the smallest x value that makes the expression undefined is x = -3.
However, the question asks for the smallest integer x where the expression is defined. The expression is defined as long as the denominator is not zero, which means x cannot take on the values that make the expression undefined.
In this case, the smallest integer x where the expression is defined would be the smallest integer greater than 2. So, the correct answer would be x = 3.
To find the smallest integer x where the expression srt(x-2)/x^2+x-6 is defined, we need to consider the restrictions on the expression.
First, let's focus on the numerator: sqrt(x-2).
To ensure the numerator is defined, we need x-2 to be greater than or equal to 0, since the square root of a negative number is undefined. Therefore, we have:
x - 2 ≥ 0
Adding 2 to both sides gives us:
x ≥ 2
Now, let's consider the denominator: x^2 + x - 6.
To ensure the denominator is defined, we need to avoid any values of x that would make the denominator equal to 0. Therefore, we have:
(x + 3)(x - 2) ≠ 0
Setting each factor to zero and solving for x gives us:
x + 3 = 0 --> x = -3
x - 2 = 0 --> x = 2
We can see that x = 2 is a value to exclude because it would make the denominator equal to 0, which would result in an undefined expression.
Therefore, the smallest integer x where the expression is defined would be the next smallest integer greater than 2, which is 3.
So, the answer is x = 3.