Answer: the solution to the inequality 5|x + 8| + 15 < 55 is -16 < x < 0.
Explanation:
To solve the inequality 5|x + 8| + 15 < 55, we need to isolate the absolute value expression and then solve for x.
First, we can start by subtracting 15 from both sides:
5|x + 8| < 40
Next, we can divide both sides by 5:
|x + 8| < 8
Now we have two cases to consider, depending on whether x + 8 is positive or negative.
Case 1: x + 8 ≥ 0
In this case, the absolute value expression can be simplified to |x + 8| = x + 8. Substituting this into the inequality, we get:
x + 8 < 8
Simplifying this expression, we get:
x < 0
So the solution for this case is x < 0.
Case 2: x + 8 < 0
In this case, the absolute value expression can be simplified to |x + 8| = -(x + 8). Substituting this into the inequality, we get:
(x + 8) < 8
Simplifying this expression, we get:
x < 16
Multiplying both sides by -1 (and reversing the inequality), we get:
x > -16
So the solution for this case is -16 < x.
Putting the solutions for both cases together, we get:
-16 < x < 0