For x+3/x-4 is less than or equal to 0, why is the interval notation [3,4)?
x+3/x-4 <= 0
now you have to be careful when multiplying by x-4.
if x-4 > 0, then x+3 <= 0 so x <= -3
But x>4, so it cannot be <= -3
if x-4 < 0 then x+3 >= 0 x >= -3
But x<4, so -3 <= x < 4 ... x in [-3,4)
To solve the inequality (x+3)/(x-4) <= 0, we need to find the values of x that make the expression on the left side less than or equal to zero.
First, let's find the critical values of x that make the expression equal to zero or result in undefined values. In this case, the expression becomes undefined when the denominator (x-4) equals zero, so x = 4 is a critical value.
Next, we need to determine the sign of the expression (x+3)/(x-4) in different intervals:
1. For x < 4: Choose a test point below 4, for example, x = 3. Substitute this value into the expression: (3+3)/(3-4) = 6/-1 = -6. The expression is negative in this interval.
2. For x > 4: Choose a test point above 4, for example, x = 5. Substitute this value into the expression: (5+3)/(5-4) = 8/1 = 8. The expression is positive in this interval.
Now, we can determine the solution based on the signs of the expression:
- It is less than or equal to zero when the expression is negative. This happens when x is less than 4.
- It is not less than or equal to zero when the expression is positive. This happens when x is greater than 4.
Thus, the solution to the inequality (x+3)/(x-4) <= 0 is x ∈ [3, 4).
The interval notation [3, 4) represents an inclusive lower bound of 3 and an exclusive upper bound of 4. It includes all values of x between 3 and 4, including 3 but not 4.