Question

problems 1-4: graph all the solutions to each inequality or equation. 1. |x| ≤ 3 2. |x + 2| > 4 |x| > 2

Explanation:

Step1: Solve \(|x|\leq3\)

The absolute - value inequality \(|x|\leq3\) is equivalent to \(-3\leq x\leq3\). On the number - line, we mark the points \(x = - 3\) and \(x = 3\) with closed circles (because the inequality includes equality) and shade the region between them.

Step2: Solve \(|x + 2|>4\)

The absolute - value inequality \(|x + 2|>4\) is equivalent to two separate inequalities: \(x+2>4\) or \(x + 2<-4\).
For \(x+2>4\), subtract 2 from both sides: \(x>4 - 2\), so \(x>2\).
For \(x + 2<-4\), subtract 2 from both sides: \(x<-4 - 2\), so \(x<-6\). On the number - line, we mark the points \(x=-6\) and \(x = 2\) with open circles (because the inequality is strict) and shade the regions \(x<-6\) and \(x>2\).

Answer:

1. For \(|x|\leq3\), the solution set is \([-3,3]\) and is represented on the number - line by shading the region between \(x=-3\) and \(x = 3\) with closed circles at \(x=-3\) and \(x = 3\).
2. For \(|x + 2|>4\), the solution set is \((-\infty,-6)\cup(2,\infty)\) and is represented on the number - line by shading the regions \(x<-6\) and \(x>2\) with open circles at \(x=-6\) and \(x = 2\).