Question

which of the following shows the graph of the inequality ( r geq -2.25 )?

Explanation:

Step1: Analyze the inequality

The inequality is \( r \geq - 2.25\). On a number line, this means we need to show all values of \( r\) that are greater than or equal to \(-2.25\). A closed circle (filled circle) is used for "greater than or equal to" or "less than or equal to" because the endpoint \(-2.25\) is included, and the arrow should point to the right (towards positive infinity) since we want values greater than \(-2.25\).

Step2: Locate \(-2.25\) on the number line

First, let's find where \(-2.25\) is on the number line. The marks are at \(-4, - 3.5,-3,-2.5,-2\). The distance between \(-2.5\) and \(-2\) is \(0.5\), so the mid - point between \(-2.5\) and \(-2\) is \(\frac{-2.5+( - 2)}{2}=\frac{-4.5}{2}=-2.25\). So \(-2.25\) is the mid - point between \(-2.5\) and \(-2\).

Step3: Check the options

• Option A: The circle is at \(-2\) (not \(-2.25\)), so it's incorrect.
• Option B: The circle is at \(-2.25\) (the mid - point between \(-2.5\) and \(-2\)), it's a closed circle (since the inequality is \(\geq\)), and the arrow points to the right (towards greater values), which matches \(r\geq - 2.25\).
• Option C: The arrow points to the left (towards smaller values), which would represent \(r\leq - 2.25\), so it's incorrect.
• Option D: The arrow points to the left and the circle is at \(-2\), so it's incorrect.

Answer:

B. The graph with a closed circle at -2.25 (mid - point between -2.5 and -2) and an arrow pointing to the right.