To solve the inequality [tex]\(2|x+1| < 4\)[/tex] for [tex]\(x\)[/tex], we will follow these steps:
1. Isolate the Absolute Value:
Begin by dividing both sides of the inequality by 2.
[tex]\[
\frac{2|x+1|}{2} < \frac{4}{2} \implies |x+1| < 2
\][/tex]
2. Create the Compound Inequality:
The inequality [tex]\( |x+1| < 2 \)[/tex] implies that the expression inside the absolute value must be between -2 and 2.
[tex]\[
-2 < x+1 < 2
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we need to subtract 1 from all parts of the compound inequality.
[tex]\[
-2 - 1 < x + 1 - 1 < 2 - 1
\][/tex]
Simplifying this, we get:
[tex]\[
-3 < x < 1
\][/tex]
This means the solution set for [tex]\(x\)[/tex] is:
[tex]\[
-3 < x < 1
\][/tex]
Graphing the Solution:
On a number line, the solution [tex]\( -3 < x < 1 \)[/tex] is represented as an open interval between -3 and 1, indicating that -3 and 1 are not included in the solution set. The graph will look like a line segment with open circles (or open dots) at -3 and 1.
Conclusion:
The solution to the inequality [tex]\(2|x+1| < 4\)[/tex] is:
[tex]\[
-3 < x < 1
\][/tex]
A number line graphically representing this solution will show an open interval between -3 and 1. The correct answer choice should include both this interval and its appropriate graphical representation.
