Question

solve the following absolute value inequality. graph the solution set on

\\|6x + 8| < -4\\

select the correct choice below and, if necessary, fill in the answer box to complete

a. the solution set is \\(\square\\).
(simplify your answer. type your answer in interval notation. use integers

b. the solution set is \\(\\{\\}\\) or \\(\varnothing\\).

Explanation:

🆕 New Concept Discovered: Solving Absolute Value Inequalities
An absolute value represents a distance, which can never be negative.

Step 1: Analyze the absolute value expression

The given inequality is:

$$ |6x + 8| < -4 $$

By definition, the absolute value of any real number expression, \( |6x + 8| \), represents its distance from zero on the number line. Because distance is always non-negative, we have:

$$ |6x + 8| \geq 0 $$

for all real numbers \( x \).

Step 2: Compare with the inequality condition

The inequality asks us to find where the absolute value is strictly less than a negative number:

$$ |6x + 8| < -4 $$

Since a non-negative number (which is \( 0 \) or greater) can never be less than a negative number (\( -4 \)), this statement is impossible.

There are no values of \( x \) that can satisfy this inequality. Therefore, there is no solution.

Answer:

B. The solution set is \(\{\}\) or \(\varnothing\).