Question

Identify the solution of the inequality −3|n 5| ≥ 24 and the graph that represents it.

Answer 1

To solve the inequality −3|n 5| ≥ 24, divide both sides by -3 and isolate the absolute value term. The solution is -8 ≤ n + 5 ≤ 8. Graph the solution by shading the region between -8 and 8 on a number line.

Step-by-step explanation:

To find the solution of the inequality −3|n 5| ≥ 24, we need to solve for n. We start by isolating the absolute value term by dividing both sides of the inequality by -3: |n 5| ≤ -8. Since an absolute value cannot be negative, we know that the absolute value of n + 5 is less than or equal to -8. Therefore, n + 5 can be any number between -8 and 8 (inclusive), or -8 ≤ n + 5 ≤ 8. To graph this solution, we plot a number line with -8 and 8 marked, and shade the region between the two points.

Answer 2

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.

1. If n + 5 is positive or zero, the inequality is -3(n + 5) ≥ 24.
2. 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: