Solve Negative two-thirds n less-than-or-equal-to 16. Which of the following must be true about the inequality and the resulting graph? Select three options. n less-than-or-equal-to negative 24 n greater-than-or-equal-to negative 24 The circle is open. The circle is closed. The arrow points right.
The Correct Answer and Explanation is :
The given inequality is:
[
-\frac{2}{3}n \leq 16
]
Step 1: Solve the inequality
To solve this inequality, the goal is to isolate (n). First, we eliminate the fraction by multiplying both sides of the inequality by (-\frac{3}{2}) (the reciprocal of (-\frac{2}{3})). However, since we are multiplying by a negative number, we need to flip the inequality.
[
n \geq 16 \times \left(-\frac{3}{2}\right)
]
[
n \geq -24
]
Step 2: Interpret the solution
The solution to the inequality is:
[
n \geq -24
]
This means that (n) can be any number greater than or equal to (-24).
Step 3: Analyze the options
- n less-than-or-equal-to negative 24: This option is incorrect because the inequality we derived shows that (n) is greater than or equal to (-24), not less than.
- n greater-than-or-equal-to negative 24: This is correct based on the solution.
- The circle is open: This option is incorrect. Since the inequality includes (n \geq -24), the circle on the graph would be closed, indicating that (-24) is included in the solution.
- The circle is closed: This is correct because (-24) is part of the solution, and the graph should have a closed circle at (-24).
- The arrow points right: This is correct because the inequality is (n \geq -24), meaning the values of (n) are greater than or equal to (-24), which is represented by an arrow pointing to the right on a number line.
Correct options:
- n greater-than-or-equal-to negative 24
- The circle is closed
- The arrow points right