Question

2 which graph represents the solution to the inequality below? show your work for full credit.
\\(\frac{x}{3} - 1 \geq -3\\)

(a) number line with open circle at -6, red to the right
(b) number line with closed circle at -6, red to the left
(c) number line with closed circle at -6, red to the right

Explanation:

Step1: Solve the inequality for x

To solve \(\frac{x}{3}-1\geq - 3\), first add 1 to both sides of the inequality.
\(\frac{x}{3}-1 + 1\geq-3 + 1\)
Simplifying both sides, we get \(\frac{x}{3}\geq - 2\).

Step2: Isolate x

Multiply both sides of the inequality \(\frac{x}{3}\geq - 2\) by 3 to isolate x.
\(3\times\frac{x}{3}\geq3\times(-2)\)
Simplifying, we have \(x\geq - 6\).

Step3: Analyze the graph

The solution \(x\geq - 6\) means we have a closed circle at \(x = - 6\) (because the inequality is "greater than or equal to") and the arrow points to the right (since we want all values greater than or equal to - 6).

Looking at the options:

• Option A: Open circle at - 6 (incorrect, since we need a closed circle for \(\geq\)) and arrow to the right.
• Option B: Closed circle at - 6 but arrow to the left (incorrect, since we need values \(\geq - 6\), which are to the right of - 6).
• Option C: Closed circle at - 6 and arrow to the right (correct, matches our solution \(x\geq - 6\)).

Answer:

C. The graph with a closed circle at -6 and an arrow pointing to the right (representing \(x\geq - 6\))