Let's break down the inequality step by step:

Given inequality:
$-|x| - 3 > -7$

Step 1: Isolate the absolute value term

Add 3 to both sides of the inequality:

$-|x| > -7 + 3$ $-|x| > -4$

Step 2: Multiply both sides by -1

When multiplying or dividing by a negative number, the inequality sign flips:

$|x| < 4$

Step 3: Solve the inequality for $x$

The inequality $|x| < 4$ means that the distance of $x$ from 0 is less than 4. So, the solution is:

$-4 < x < 4$

Step 4: Consider the domain $D = \{\text{Integers}\}$

Since the problem specifies that the domain is the set of integers, we need to find all integer values of $x$ that satisfy this inequality.

The integers that lie between -4 and 4 (not including -4 and 4 themselves) are: $x = -3, -2, -1, 0, 1, 2, 3$

Step 5: Graph on a number line

You will plot these values as open circles at $x = -3, -2, -1, 0, 1, 2, 3$, since the inequality is strict ($<$, not $\leq$).

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Let me know if you would like me to graph it for you or if you'd like further clarification!

Here are 5 questions to expand on this topic:

1. What happens if the inequality is $-|x| - 3 \geq -7$?
2. How would the solution change if the domain were real numbers instead of integers?
3. Can we graph absolute value inequalities on a number line even if the domain is continuous?
4. How do we solve inequalities involving absolute value with a negative right-hand side?
5. What does it mean geometrically to solve absolute value inequalities?

Tip: Always remember to flip the inequality sign when multiplying or dividing by a negative number!