Solve the following absolute value inequality:

[tex]\[
\frac{4|x+9|}{5} \ \textless \ 8
\][/tex]

Determine the values of [tex]\( x \)[/tex].



Answer :

To solve the absolute value inequality:

[tex]\[ \frac{4|x+9|}{5} < 8 \][/tex]

we can follow these steps:

1. Isolate the absolute value:

[tex]\[ \frac{4|x+9|}{5} < 8 \][/tex]

First, clear the fraction by multiplying both sides by [tex]\(5\)[/tex]:

[tex]\[ 4|x+9| < 40 \][/tex]

Next, divide both sides by [tex]\(4\)[/tex]:

[tex]\[ |x+9| < 10 \][/tex]

2. Split the absolute value into two inequalities:

The expression [tex]\(|x+9| < 10\)[/tex] indicates that the value inside the absolute value, [tex]\(x+9\)[/tex], must be between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex]. Therefore, we can write:

[tex]\[ -10 < x + 9 < 10 \][/tex]

3. Solve the compound inequality:

To isolate [tex]\(x\)[/tex], subtract [tex]\(9\)[/tex] from all parts of the inequality:

[tex]\[ -10 - 9 < x + 9 - 9 < 10 - 9 \][/tex]

Simplifying the inequality, we get:

[tex]\[ -19 < x < 1 \][/tex]

Therefore, the solution to the inequality [tex]\(\frac{4|x+9|}{5} < 8\)[/tex] is:

[tex]\[ -19 < x < 1 \][/tex]

In other words,
[tex]\[ x > -19 \quad \text{and} \quad x < 1 \][/tex]