Question

solve the inequality and graph the solution.
4(w + 6) > 8
plot the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.

Explanation:

Step1: Expand the left - hand side

Use the distributive property \(a(b + c)=ab+ac\). Here \(a = 4\), \(b=w\), \(c = 6\).
\(4(w + 6)=4w+24\). So the inequality becomes \(4w + 24>8\).

Step2: Subtract 24 from both sides

To isolate the term with \(w\), use the subtraction property of inequalities.
\(4w+24-24>8 - 24\).
\(4w>-16\).

Step3: Divide both sides by 4

Use the division property of inequalities. Since \(4>0\), the direction of the inequality sign does not change.
\(\frac{4w}{4}>\frac{-16}{4}\).
\(w>-4\).

To graph \(w > - 4\):

• The endpoint is \(w=-4\). Since the inequality is \(>\) (not \(\geq\)), the endpoint at \(w = - 4\) is open.
• The solution set is all real numbers greater than \(-4\), so we draw a ray starting at \(w=-4\) and going to the right.

Answer:

The solution of the inequality \(4(w + 6)>8\) is \(w>-4\). On the number - line, there is an open circle at \(w = - 4\) and a ray extending to the right.