To solve the inequality -3|n + 5| ≥ 24, we consider two cases for n + 5 being positive or negative, and solve the resulting inequalities to obtain n ≤ -13 or n ≥ 3. To graph this, we use a number line with closed circles at -13 and 3 and shade outward from these points.
To identify the solution of the inequality −3|n + 5| ≥ 24, we must first address the absolute value. This gives us two scenarios because the value inside the absolute value can be both positive and negative.
- If n + 5 is positive or zero, the inequality is -3(n + 5) ≥ 24.
- If n + 5 is negative, the inequality is -3(-n - 5) ≥ 24 which simplifies to 3(n + 5) ≥ 24.
For the first scenario, we divide both sides by -3 (remembering to reverse the inequality because we are dividing by a negative number), resulting in:
n + 5 ≤ -8
Then, subtracting 5 from both sides:
n ≤ -13
For the second scenario, we divide both sides by 3:
n + 5 ≥ 8
Subtracting 5 from both sides gives us:
n ≥ 3
Combining these two results, we find that n ≤ -13 or n ≥ 3. To graph this solution, draw a number line, plot points at -13 and 3, use a closed circle to include the points -13 and 3, and shade the line to the left of -13 and to the right of 3.
The graph is given below: