Question

graph the solution to the inequality on the number line.
|u + 4| ≥ 3

Explanation:

Step1: Solve the absolute - value inequality

For the absolute - value inequality \(|a|\geq b\) (where \(b > 0\)), it is equivalent to \(a\geq b\) or \(a\leq - b\).
In the inequality \(|u + 4|\geq3\), we have two cases:
Case 1: \(u+4\geq3\)
Subtract 4 from both sides of the inequality: \(u+4 - 4\geq3 - 4\), which simplifies to \(u\geq - 1\).
Case 2: \(u + 4\leq-3\)
Subtract 4 from both sides of the inequality: \(u+4 - 4\leq-3 - 4\), which simplifies to \(u\leq - 7\).

Step2: Graph the solution on the number line

• For \(u\geq - 1\): We use a closed dot (because the inequality is "greater than or equal to") at \(u=-1\) and draw an arrow to the right (towards positive infinity).
• For \(u\leq - 7\): We use a closed dot (because the inequality is "less than or equal to") at \(u = - 7\) and draw an arrow to the left (towards negative infinity).

To graph on the given number line:

• At \(u=-7\), place a closed dot (the blue dot) and draw an arrow to the left (using the arrow tool).
• At \(u=-1\), place a closed dot (the blue dot) and draw an arrow to the right (using the arrow tool).

Answer:

The solution to the inequality \(|u + 4|\geq3\) is \(u\leq - 7\) or \(u\geq - 1\). On the number line, we mark a closed dot at \(-7\) and draw an arrow to the left, and mark a closed dot at \(-1\) and draw an arrow to the right.