Question

question 9 of 10
solve the inequality for x and identify the graph of its solution.
|x + 3|>2
choose the answer that gives both the correct solution and the correct graph.
a. solution: x>1 and x<5
-2 -1 0 1 2 3 4 5 6 7 8
b. solution: x>-5 and x<-1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
c. solution: x<-5 or x>-1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
d. solution: x<-5 or x>-1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

Explanation:

Step1: Recall absolute - value inequality rule

For \(|u|>a\) (\(a > 0\)), the solution is \(u<-a\) or \(u > a\). Here \(u=x + 3\) and \(a = 2\), so \(x+3<-2\) or \(x + 3>2\).

Step2: Solve \(x+3<-2\)

Subtract 3 from both sides: \(x+3-3<-2 - 3\), which gives \(x<-5\).

Step3: Solve \(x + 3>2\)

Subtract 3 from both sides: \(x+3-3>2-3\), which gives \(x>-1\).

Step4: Analyze the graph

The solution \(x<-5\) or \(x>-1\) is represented by two rays on the number - line. One ray goes to the left of \(-5\) (not including \(-5\)) and the other goes to the right of \(-1\) (not including \(-1\)).

Answer:

D. Solution: \(x < - 5\) or \(x>-1\) (with the correct graph of two rays, one to the left of \(-5\) and one to the right of \(-1\) with open - circles at \(-5\) and \(-1\))