To solve the inequality [tex]\( |x+5| < 2x + 16 \)[/tex], let's break it down into steps to find the solution:
1. Understand the Absolute Value Inequality:
The inequality involves an absolute value expression, [tex]\( |x+5| \)[/tex]. Recall that [tex]\( |a| < b \)[/tex] means [tex]\(-b < a < b\)[/tex].
2. Set Up the Inequalities:
Given [tex]\( |x+5| < 2x + 16 \)[/tex], we can split this into two separate inequalities:
[tex]\[
-(x+5) < 2x + 16 \quad \text{and} \quad x+5 < 2x + 16.
\][/tex]
3. Solve Each Inequality Separately:
- For the first inequality:
[tex]\[
-(x+5) < 2x + 16
\][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
-x - 5 < 2x + 16
\][/tex]
Add [tex]\( x \)[/tex] to both sides:
[tex]\[
-5 < 3x + 16
\][/tex]
Subtract 16 from both sides:
[tex]\[
-21 < 3x
\][/tex]
Divide by 3:
[tex]\[
-7 < x \quad \text{or} \quad x > -7.
\][/tex]
- For the second inequality:
[tex]\[
x + 5 < 2x + 16
\][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
5 < x + 16
\][/tex]
Subtract 16 from both sides:
[tex]\[
-11 < x \quad \text{or} \quad x > -11.
\][/tex]
4. Combine the Results:
Both inequalities need to be satisfied simultaneously. Therefore, we take the intersection of the solutions:
[tex]\[
x > -7 \quad \text{and} \quad x > -11.
\][/tex]
The more restrictive condition is [tex]\( x > -7 \)[/tex], as it is a stronger constraint.
Hence, the solution to the inequality [tex]\( |x+5| < 2x + 16 \)[/tex] is:
[tex]\[
\boxed{x > -7}
\][/tex]
So, the correct answer is (A) [tex]\( x > -7 \)[/tex].
