To solve the inequality \( 1 - (x + 3) \geq 4 + 2x \) and express the solution in interval notation, let's follow these steps:

### Step 1: Simplify the Inequality
Distribute the negative sign:
\[
1 - x - 3 \geq 4 + 2x
\]
Simplify the left side:
\[
- x - 2 \geq 4 + 2x
\]

### Step 2: Move Terms Involving \( x \) to One Side
Add \( x \) to both sides:
\[
-2 \geq 4 + 3x
\]

### Step 3: Isolate \( x \)
Subtract 4 from both sides:
\[
-6 \geq 3x
\]

### Step 4: Solve for \( x \)
Divide both sides by 3:
\[
-2 \geq x
\]
or equivalently,
\[
x \leq -2
\]

### Step 5: Write the Solution in Interval Notation
The solution is all \( x \) such that \( x \leq -2 \). In interval notation, this is:
\[
(-\infty, -2]
\]

### Final Answer:
The solution in interval notation is \( \boxed{(-\infty, -2]} \).

Would you like further details or explanations on any part of this process?

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Here are 5 related questions you might consider:

1. How would the interval change if the inequality was strict (i.e., \( 1 - (x + 3) > 4 + 2x \))?
2. Can you solve the inequality \( 2(x - 1) \leq 3x + 5 \) and express it in interval notation?
3. What would the solution look like if we added absolute value to the inequality, such as \( |x + 3| \geq 4 + 2x \)?
4. How does the interval notation change if \( x \) were greater than or equal to \(-2\) instead?
5. Can you solve a compound inequality, like \( 1 - (x + 3) \geq 4 + 2x \) and \( x + 5 \leq 3 \), and express the solution in interval notation?

**Tip:** Always double-check your inequality when solving for \( x \); a small mistake in distributing or moving terms can lead to an incorrect interval.

Learn how to solve the inequality 1 - (x + 3) ≥ 4 + 2x step-by-step and express the solution in interval notation. This problem covers concepts in inequalities and interval notation, suitable for Grades 9-12.

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