Question

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Explanation:

Step1: Analyze the compound inequality

The compound inequality is \( x < 8 \) and \( x \geq - 5 \). We need to find the values of \( x \) that satisfy both inequalities simultaneously.

Step2: Combine the inequalities

To express the solution in set - builder notation, we combine the two inequalities. The values of \( x \) must be greater than or equal to \(-5\) and less than \( 8 \). So the inequality is \( - 5\leq x<8 \).

Step3: Determine the correct option for set - builder notation

For option A, we fill in the box with \( - 5\leq x<8 \). Option B is incorrect because the solution is not all real numbers (it is restricted between \(-5\) and \(8\)). Option C is incorrect because there are values of \( x \) (e.g., \( 0\)) that satisfy both inequalities, so the solution set is not empty.

Step4: Analyze the graph of the solution

The solution set \( - 5\leq x<8 \) means that we have a closed circle at \( x=-5\) (because \( x\) can be equal to \(-5\)) and an open circle at \( x = 8\) (because \( x\) is less than \( 8\)) and the line segment between them. Looking at the graphs, graph C (assuming graph C has a closed circle at \(-5\), an open circle at \(8\), and the line connecting them) is the correct graph. But since we are mainly focused on the set - builder notation part first, we will first handle the set - builder notation.

Answer:

(for set - builder notation):
A. The solution is \(\{x\mid - 5\leq x<8\}\)

(For the graph part, assuming the graphs are labeled as in the problem, the correct graph is the one with a closed dot at \(-5\), an open dot at \(8\), and the line segment between them, which is option C (depending on the actual graph's visual representation). But since the main question here was about the set - builder notation first, we provided that. If we consider the graph part, we would analyze each graph:

• Graph A: Seems to have an open circle at \(8\) and going to the right, which is \(x > 8\), incorrect.
• Graph B: No marking at \(-5\) or \(8\), incorrect.
• Graph C: Closed at \(-5\), open at \(8\), line in between, correct.
• Graph D: No restrictions, all real numbers, incorrect.
• Graph E: Closed at some negative number less than \(-5\), incorrect.
• Graph F: Two separate rays, incorrect.

So the graph of the solution is option C.)