Question

solve the inequality and graph the solution.
4(d + 4) ≤ -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 \(a = 4\), \(b=d\), \(c = 4\).
\(4(d + 4)=4d+16\). So the inequality becomes \(4d+16\leq - 8\).

Step2: Subtract 16 from both sides

Subtract 16 from each side of the inequality \(4d+16-16\leq - 8 - 16\).
\(4d\leq-24\).

Step3: Divide both sides by 4

Divide each side of the inequality \(4d\leq-24\) by 4. Since \(4>0\), the direction of the inequality sign does not change.
\(d\leq\frac{-24}{4}\), so \(d\leq - 6\).

To graph the solution:

• The endpoint is at \(d=-6\). Since the inequality is \(\leq\), the endpoint is closed (a filled - in circle).
• The graph is a ray that extends to the left (towards more negative values) from \(d = - 6\) on the number line.

Answer:

The solution of the inequality \(4(d + 4)\leq-8\) is \(d\leq - 6\). On the number line, we have a closed circle at \(-6\) and a ray extending to the left.