To solve the inequality $2|x + 9| + 2 \leq 10$, we will first isolate the absolute value term and then consider the two cases that arise from it.

Step 1: Isolate the Absolute Value

Subtract $2$ from both sides: $2|x + 9| \leq 8$

Now, divide both sides by $2$: $|x + 9| \leq 4$

Step 2: Remove the Absolute Value

The inequality $|x + 9| \leq 4$ implies that: $-4 \leq x + 9 \leq 4$

Step 3: Solve for $x$

Now we can split this compound inequality into two parts and solve each part:

1. Left Inequality: $-4 \leq x + 9$ Subtract $9$ from both sides: $x \geq -13$

2. Right Inequality: $x + 9 \leq 4$ Subtract $9$ from both sides: $x \leq -5$

Step 4: Combine the Results

The solution to the inequality is: $-13 \leq x \leq -5$

Solution

In interval notation, the solution set is: $[-13, -5]$

This means any $x$ within this range satisfies the original inequality $2|x + 9| + 2 \leq 10$.

Would you like more details, or do you have any questions about these steps?

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Here are five related questions to deepen your understanding:

1. How would you solve an inequality with a more complex absolute value expression, like $3|2x - 1| - 7 \leq 11$?
2. What happens if the inequality is strict, such as $2|x + 9| + 2 < 10$?
3. Can you explain why we need to consider two cases when solving absolute value inequalities?
4. How would the solution change if the inequality were $2|x + 9| + 2 \geq 10$?
5. What are some practical applications of absolute value inequalities?

Tip: Always start by isolating the absolute value term when solving absolute value inequalities. This will simplify the inequality and help you identify the cases to consider.