Question

calculator active directions for the calculator active items: - calculators may be used on calculator active items. - read each problem carefully. - choose the best answer from the choices given. - fractions in some answer choices may have been simplified. check each answer choice to see if this has been done. - diagrams used in the test may not be drawn to scale. - at this time, you can check your answers for calculator active items only. which graph correctly displays the solutions to the inequality $-\frac{1}{2}n - 81 \leq -75$? four number line graphs with options, first has open circle at -12 and arrow left, second has closed circle at -12 and arrow right, third has open circle at -9 and arrow right, fourth has closed circle at -9 and arrow left

Explanation:

Step1: Solve the inequality

Start with the inequality \(-\frac{1}{2}n - 81 \leq -75\). First, add 81 to both sides:
\(-\frac{1}{2}n - 81 + 81 \leq -75 + 81\)
Simplify to get \(-\frac{1}{2}n \leq 6\).
Then, multiply both sides by -2. Remember, when multiplying or dividing an inequality by a negative number, the inequality sign flips. So:
\((-2)\times(-\frac{1}{2}n) \geq 6\times(-2)\)
Simplify to find \(n \geq -12\).

Step2: Analyze the number line graphs

We need a number line with a closed circle (since the inequality is "less than or equal to" after solving, but wait, no—wait, after solving, we have \(n \geq -12\), so the circle at -12 should be closed (because the inequality includes equality, \(n = -12\) is a solution), and the arrow should point to the right (since \(n\) is greater than or equal to -12).

Looking at the options:

• First graph: Open circle at -12, arrow left. Incorrect.
• Second graph: Closed circle at -12, arrow right. Correct.
• Third graph: Open circle at -9, arrow right. Incorrect.
• Fourth graph: Closed circle at -9, arrow left. Incorrect.

Answer:

The second graph (the one with closed circle at -12 and arrow pointing right)