Certainly! Let's solve the given absolute value equation step by step.
[tex]\[
\frac{10|x+8|}{5} = 24
\][/tex]
First, let's simplify the left side of the equation:
[tex]\[
\frac{10|x+8|}{5} = 2|x+8|
\][/tex]
So the equation becomes:
[tex]\[
2|x+8| = 24
\][/tex]
Now, divide both sides of the equation by 2:
[tex]\[
|x+8| = \frac{24}{2}
\][/tex]
[tex]\[
|x+8| = 12
\][/tex]
Next, we need to consider the definition of the absolute value function, which tells us that:
[tex]\[
|A| = B \implies A = B \quad \text{or} \quad A = -B
\][/tex]
In our case:
[tex]\[
x + 8 = 12 \quad \text{or} \quad x + 8 = -12
\][/tex]
Let's solve each of these equations separately:
1. For the first equation:
[tex]\[
x + 8 = 12
\][/tex]
Subtract 8 from both sides:
[tex]\[
x = 12 - 8
\][/tex]
[tex]\[
x = 4
\][/tex]
2. For the second equation:
[tex]\[
x + 8 = -12
\][/tex]
Subtract 8 from both sides:
[tex]\[
x = -12 - 8
\][/tex]
[tex]\[
x = -20
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x = 4 \quad \text{or} \quad x = -20
\][/tex]
Hence:
[tex]\[
x = 4
\quad \text{and} \quad
x = -20
\][/tex]
So in conclusion, the values of [tex]\( x \)[/tex] that satisfy the given equation are [tex]\( 4 \)[/tex] and [tex]\( -20 \)[/tex].
