Question

solve and graph the compound inequality. 2x + 1 ≤ 7 and -2x + 1 ≤ 9

Explanation:

Step1: Solve the first - inequality

Solve \(2x + 1\leq7\). Subtract 1 from both sides: \(2x\leq7 - 1\), so \(2x\leq6\). Then divide both sides by 2: \(x\leq3\).

Step2: Solve the second - inequality

Solve \(-2x + 1\leq9\). Subtract 1 from both sides: \(-2x\leq9 - 1\), so \(-2x\leq8\). Divide both sides by - 2 and reverse the inequality sign: \(x\geq - 4\).

Step3: Combine the solutions

The solution of the compound inequality \(2x + 1\leq7\) and \(-2x + 1\leq9\) is \(-4\leq x\leq3\).

Step4: Analyze the graph

On a number - line, for \(x\geq - 4\), there is a closed circle at \(x=-4\) (because the inequality is \(\geq\)) and the line goes to the right. For \(x\leq3\), there is a closed circle at \(x = 3\) (because the inequality is \(\leq\)) and the line goes to the left.

Answer:

The correct graph is the one with a closed circle at \(x=-4\), a closed circle at \(x = 3\), and the line segment between them. Without seeing the specific options clearly labeled in terms of which one has the correct closed - circle and line - segment representation, if we assume the standard number - line graphs, the graph that shows a closed dot at \(-4\), a closed dot at \(3\), and the line connecting them represents the solution set \(-4\leq x\leq3\).