Question

1. graph the solution to the inequality.

$-\frac{1}{3}x\geq2$ or $2(x + 1)>8$

Explanation:

Step1: Solve the first inequality \(-\frac{1}{3}x\geq2\)

Multiply both sides by \(-3\) (and reverse the inequality sign because we are multiplying by a negative number).
\(x\leq2\times(- 3)\), so \(x\leq - 6\).

Step2: Solve the second inequality \(2(x + 1)>8\)

First, divide both sides by \(2\): \(x + 1>4\).
Then subtract \(1\) from both sides: \(x>4 - 1\), so \(x>3\).

Step3: Graph the solutions

For \(x\leq - 6\), since the inequality is non - strict (\(\leq\)), we use a closed circle at \(x=-6\) and shade to the left (towards more negative numbers). For \(x > 3\), since the inequality is strict (\(>\)), we use an open circle at \(x = 3\) and shade to the right (towards more positive numbers).

Answer:

The solution is \(x\leq - 6\) or \(x>3\). On the number - line, for \(x\leq - 6\), we have a closed circle at \(x = - 6\) and an arrow to the left. For \(x>3\), we have an open circle at \(x = 3\) and an arrow to the right.