We are tasked with solving the inequality:

$5|x + 7| + 1 \leq 36$

Step-by-step solution:

1. Isolate the absolute value term:

   Subtract 1 from both sides: $5|x + 7| \leq 35$

2. Divide by 5:

   $|x + 7| \leq 7$

3. Solve the absolute value inequality:

   Recall that for $|A| \leq B$, where $B \geq 0$, the inequality translates to: $-B \leq A \leq B$ Applying this to $|x + 7| \leq 7$, we have: $-7 \leq x + 7 \leq 7$

4. Solve for $x$:

   Subtract 7 from all parts of the inequality: $-7 - 7 \leq x \leq 7 - 7$ Simplifying: $-14 \leq x \leq 0$

Final answer:

The solution to the inequality is:

$x \in [-14, 0]$

Would you like more details, or do you have any further questions?

5 Related Questions:

1. How do you solve absolute value inequalities with greater than signs?
2. What happens when there's a negative coefficient in front of the absolute value?
3. Can you graph this inequality on a number line?
4. How do absolute value functions behave differently from regular linear equations?
5. How can this inequality be solved by graphing instead of algebraically?

Tip:

Always isolate the absolute value term first when solving inequalities that involve absolute values.