State True or False: The solution set of `|x+2|le5` is `[-7,3]`.

Answer

To determine whether the statement "The solution set of \(|x+2| \leq 5\) is \([-7, 3]\)" is true or false, we will solve the inequality step by step.

### Step 1: Understand the Absolute Value Inequality
The inequality \(|x + 2| \leq 5\) means that the expression inside the absolute value, \(x + 2\), must lie within the range of \(-5\) to \(5\).

### Step 2: Set Up the Compound Inequality
From the definition of absolute value, we can rewrite the inequality as:
\[ -5 \leq x + 2 \leq 5 \]

### Step 3: Solve the Left Side of the Inequality
Start by solving the left part of the compound inequality:
\[ -5 \leq x + 2 \]
Subtract \(2\) from both sides:
\[ -5 - 2 \leq x \implies -7 \leq x \implies x \geq -7 \]

### Step 4: Solve the Right Side of the Inequality
Now, solve the right part of the compound inequality:
\[ x + 2 \leq 5 \]
Subtract \(2\) from both sides:
\[ x \leq 5 - 2 \implies x \leq 3 \]

### Step 5: Combine the Results
Now we combine the results from both sides:
\[ -7 \leq x \leq 3 \]
This can be written in interval notation as:
\[ x \in [-7, 3] \]

### Conclusion
The solution set of \(|x + 2| \leq 5\) is indeed \([-7, 3]\). Therefore, the statement is **True**.

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