Question

question 9 of 10
solve the inequality for x and identify the graph of its solution.
|x + 2|>2
choose the answer that gives both the correct solution and the correct graph
a. solution: x < - 4 or x>0

• 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3

b. solution: x> - 4 and x < 0

• 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3

c. solution: x≥ - 4 and x≤0

• 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3

d. solution: x < 0 or x>4

• 3 - 2 - 1 0 1 2 3 4 5 6 7

Explanation:

Step1: Recall absolute - value inequality rule

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

Step2: Solve the left - hand side of the compound inequality

Subtract 2 from both sides of \(-2

Step3: Solve the right - hand side of the compound inequality

Subtract 2 from both sides of \(x + 2<2\): \(x+2-2<2 - 2\), which simplifies to \(x<0\).

Step4: Interpret the solution in interval notation and graph

The solution of the inequality \(|x + 2|<2\) is \(x>-4\) and \(x<0\). On a number line, we have an open - circle at \(x=-4\) and \(x = 0\) and the line segment between them is shaded.

Answer:

B. Solution: \(x>-4\) and \(x<0\)