To solve the inequality [tex]\(-\frac{2}{3} n \leq 16\)[/tex], follow these steps:
1. Isolate [tex]\( n \)[/tex]:
- To isolate [tex]\( n \)[/tex], multiply both sides of the inequality by the reciprocal of [tex]\(-\frac{2}{3}\)[/tex], which is [tex]\(-\frac{3}{2}\)[/tex].
- When multiplying or dividing both sides by a negative number, remember to reverse the inequality sign.
2. Solve for [tex]\( n \)[/tex]:
- Multiply both sides of the inequality [tex]\( -\frac{2}{3} n \leq 16 \)[/tex] by [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[
n \geq 16 \times -\frac{3}{2}
\][/tex]
- Calculate the right-hand side:
[tex]\[
n \geq -24
\][/tex]
So, the solution to the inequality [tex]\( -\frac{2}{3} n \leq 16 \)[/tex] is [tex]\( n \geq -24 \)[/tex].
Next, let's evaluate the given statements about the inequality and the resulting graph:
- [tex]\( n \leq -24 \)[/tex]: This statement is false because the solution indicates that [tex]\( n \)[/tex] is greater than or equal to [tex]\(-24\)[/tex].
- [tex]\( n \geq -24 \)[/tex]: This statement is true because we've determined that [tex]\( n \)[/tex] must be greater than or equal to [tex]\(-24\)[/tex].
- The circle is open: In inequalities like [tex]\( n \geq -24 \)[/tex], the circle on the number line is closed at [tex]\(-24\)[/tex] because the inequality includes the equal sign ( [tex]\(\geq\)[/tex] ).
- The circle is closed: This statement is true because the inequality [tex]\( n \geq -24 \)[/tex] includes the [tex]\( -24 \)[/tex] itself, indicating a closed circle at that point on the number line.
- The arrow points right: This statement is false because the solution [tex]\( n \geq -24 \)[/tex] indicates that [tex]\( n \)[/tex] can be any number greater than or equal to [tex]\(-24\)[/tex], which means the arrow points to the right to indicate increasing values.
Thus, the correct statements are:
1. [tex]\( n \geq -24 \)[/tex]
2. The circle is closed
3. The arrow points right
