Question

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

Explanation:

Step1: Recall absolute - value inequality rule

For \(|a|\lt b\) (\(b>0\)), it is equivalent to \(-b\lt a\lt b\). Here \(a = x + 1\) and \(b = 2\), so \(-2\lt x + 1\lt2\).

Step2: Solve the compound - inequality

Subtract 1 from all parts of the compound - inequality: \(-2-1\lt x+1 - 1\lt2 - 1\), which simplifies to \(-3\lt x\lt1\).
The graph of the solution \(x>-3\) and \(x < 1\) is a number line with open circles at \(x=-3\) and \(x = 1\) and the line segment between them shaded.

Answer:

B. Solution: \(x>-3\) and \(x<1\)
(Graph: A number line with open - circles at \(x=-3\) and \(x = 1\) and the line segment between them shaded)