Question

the solution set is (-infty, -1) cup 7, infty)
part: 1 / 2
part 2 of 2
graph the solution set for ( 4 leq |n - 3| ).
number line from -9 to 9 with tick marks, and a toolbox with ( ), , and arrows

Explanation:

Step1: Solve the absolute - value inequality

Given the inequality \(4\leq|n - 3|\). By the definition of absolute value, if \(|x|\geq a\) (\(a>0\)), then \(x\leq - a\) or \(x\geq a\). Here \(x=n - 3\) and \(a = 4\), so we have two cases:
Case 1: \(n-3\leq - 4\)
Add 3 to both sides of the inequality: \(n-3 + 3\leq-4 + 3\), which simplifies to \(n\leq - 1\).
Case 2: \(n - 3\geq4\)
Add 3 to both sides of the inequality: \(n-3 + 3\geq4 + 3\), which simplifies to \(n\geq7\).

Step2: Graph the solution set

• For \(n\leq - 1\): On the number line, we use a closed dot at \(n=-1\) (because the inequality is "less than or equal to") and draw an arrow to the left (towards negative infinity).
• For \(n\geq7\): On the number line, we use a closed dot at \(n = 7\) (because the inequality is "greater than or equal to") and draw an arrow to the right (towards positive infinity).

Answer:

To graph the solution set:

• Place a closed circle (the "[" or "]" - like symbol for inclusive) at \(-1\) and draw an arrow to the left.
• Place a closed circle at \(7\) and draw an arrow to the right.