Question

graph the solution to the inequality on the number line.
$|-3 - 3x| \geq 6$
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

Explanation:

Step1: Solve the absolute - value inequality

For \(|a|\geq b\) (\(b\geq0\)), the solution is \(a\geq b\) or \(a\leq - b\).
Here \(a=-3 - 3x\) and \(b = 6\). So we have two inequalities:

• Inequality 1: \(-3-3x\geq6\)

Add \(3\) to both sides: \(-3x\geq6 + 3\), i.e., \(-3x\geq9\).
Divide both sides by \(-3\) (and reverse the inequality sign): \(x\leq-3\).

• Inequality 2: \(-3-3x\leq - 6\)

Add \(3\) to both sides: \(-3x\leq-6 + 3\), i.e., \(-3x\leq-3\).
Divide both sides by \(-3\) (and reverse the inequality sign): \(x\geq1\).

Step2: Graph the solution on the number - line

• For \(x\leq-3\), we use a closed circle at \(x = - 3\) (because the inequality is \(\leq\)) and shade to the left.
• For \(x\geq1\), we use a closed circle at \(x = 1\) (because the inequality is \(\geq\)) and shade to the right.

Answer:

The solution of the inequality \(| - 3-3x|\geq6\) is \(x\leq-3\) or \(x\geq1\). On the number - line, we have a closed circle at \(x=-3\) with shading to the left and a closed circle at \(x = 1\) with shading to the right.