Question

When solving the absolute value inequality 2|x+5|<8 , it is necessary to first isolate the absolute value. Which absolute value rule must be applied, or which special case is this an example of?(1 point)
Responses

"or rule"
"or rule"

"and rule"
"and rule"

No solution
No solution

All real numbers are solutions.
All real numbers are solutions.

Answer 1

To solve the absolute value inequality \( 2|x+5| < 8 \), the first step is to isolate the absolute value by dividing both sides by 2, which gives us:

\[
|x + 5| < 4
\]

This leads us to the "and rule" for absolute values, which states that if \( |A| < B \) (where \( B > 0 \)), then \(-B < A < B\).

In this case, we can rewrite the inequality as:

\[
-4 < x + 5 < 4
\]

Now, we can solve this double inequality:

1. Subtract 5 from all parts:
\[
-4 - 5 < x < 4 - 5
\]
\[
-9 < x < -1
\]

Therefore, the appropriate response is:

**"and rule"**