Question

graph the solution to the inequality on the number line.
|2w - 6| ≥ 14

Explanation:

Step 1: Solve the absolute - value inequality

Recall the property of absolute - value inequalities: If \(|x|\geq a\) (where \(a>0\)), then \(x\leq - a\) or \(x\geq a\).
For the inequality \(|2w - 6|\geq14\), we can apply this property. Let \(x = 2w-6\) and \(a = 14\). So we get two separate inequalities:
\(2w-6\leq - 14\) or \(2w - 6\geq14\)

Step 2: Solve \(2w-6\leq - 14\)

Add \(6\) to both sides of the inequality:
\(2w-6 + 6\leq - 14+6\)
\(2w\leq - 8\)
Divide both sides by \(2\):
\(\frac{2w}{2}\leq\frac{-8}{2}\)
\(w\leq - 4\)

Step 3: Solve \(2w - 6\geq14\)

Add \(6\) to both sides of the inequality:
\(2w-6 + 6\geq14 + 6\)
\(2w\geq20\)
Divide both sides by \(2\):
\(\frac{2w}{2}\geq\frac{20}{2}\)
\(w\geq10\)

Answer:

The solution to the inequality \(|2w - 6|\geq14\) is \(w\leq - 4\) or \(w\geq10\). To graph this on the number line:

• For \(w\leq - 4\), we draw a closed circle at \(-4\) (since the inequality is "less than or equal to") and shade to the left of \(-4\).
• For \(w\geq10\), we draw a closed circle at \(10\) (since the inequality is "greater than or equal to") and shade to the right of \(10\).