Write the solution to the given inequality in interval notation.
[-3,-1)
(-3,-1]
(-∞,-3) U [-1,∞)
(-∞,-3] U (-1,∞)
To write the solution to the given inequality in interval notation, we need to consider each interval and combine them if necessary.
The given inequality is [-3, -1).
To write this in interval notation, we use square brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. The interval goes from -3 to -1, including -3 but excluding -1. Therefore, we write this interval as (-3, -1).
The second inequality is (-3, -1].
Using the same rules as above, this interval goes from -3 to -1, excluding -3 but including -1. Therefore, we write this interval as (-3, -1].
The third inequality is (-∞, -3) U [-1, ∞).
The symbol (-∞) represents all real numbers that are less than -3, and the symbol (∞) represents all real numbers greater than -1.
To express this in interval notation, we write the first part as (-∞, -3), which includes all real numbers less than -3.
The second part is represented by the interval [-1, ∞), which includes all real numbers from -1 onwards.
Therefore, we combine these two intervals using the union symbol U to get (-∞, -3) U [-1, ∞).
The fourth inequality is (-∞, -3] U (-1, ∞).
This can be written in interval notation as (-∞, -3] U (-1, ∞), which represents all real numbers less than or equal to -3 combined with all real numbers between -1 and ∞, excluding -1.
In summary:
(-3, -1)
(-3, -1]
(-∞, -3) U [-1, ∞)
(-∞, -3] U (-1, ∞)
To write the solution to the given inequality in interval notation, we need to understand the notation used.
In interval notation, we use square brackets [ ] to indicate that the endpoint is included in the interval, and parentheses ( ) to indicate that the endpoint is not included in the interval. We also use infinity symbol (∞) to represent positive or negative infinity.
Let's analyze each option and determine the correct one:
1. [-3, -1):
This notation means that the interval includes -3 but does not include -1. So, the solution to the inequality is all the numbers greater than or equal to -3 and less than -1. In interval notation, this would be [-3, -1).
2. (-3, -1]:
This notation means that the interval includes -1 but does not include -3. So, the solution to the inequality is all the numbers greater than -3 and less than or equal to -1. In interval notation, this would be (-3, -1].
3. (-∞, -3) U [-1, ∞):
This notation implies two separate intervals connected by a union operator (U). The first interval is (-∞, -3), which includes all the numbers less than -3. The second interval is [-1, ∞), which includes all the numbers greater than or equal to -1. So, the solution to the inequality is all the numbers less than -3 or greater than or equal to -1. In interval notation, this would be (-∞, -3) U [-1, ∞).
4. (-∞, -3] U (-1, ∞):
Similar to the previous option, this notation also implies two separate intervals connected by a union operator (U). The first interval is (-∞, -3], which includes all the numbers less than or equal to -3. The second interval is (-1, ∞), which includes all the numbers greater than -1. So, the solution to the inequality is all the numbers less than or equal to -3 or greater than -1. In interval notation, this would be (-∞, -3] U (-1, ∞).
Therefore, the correct solution to the given inequality in interval notation is (-∞, -3] U (-1, ∞).