number line below.

To solve the inequality \( x - 3 \geq -4 + x \), we first simplify it:

1. Subtract \( x \) from both sides: \[ x - x - 3 \geq -4 \] This simplifies to: \[ -3 \geq -4 \]

2. This inequality is always true since \(-3\) is greater than \(-4\).

Since the inequality holds for all values of \( x \), the solution can be expressed in inequality notation as: \[ \text{Inequality Notation: } (-\infty, \infty) \]

For the number line, this means that all numbers are included. You can represent it as:

Number Line: -12 —————— -10 —————— -8 —————— -6 —————— -4 —————— -2 —————— 0 —————— 2 —————— 4 —————— 6 —————— 8 —————— 10 —————— 12

With shading or marking along the entire line to indicate that every point is part of the solution set.

So, the final answer is:

Inequality Notation: \((- \infty, \infty)\)
Number Line: [shaded/marked for all values]

For the inequality \(x - 3 \geq -4 + x\), we found that this is true for all values of \(x\).

Since the inequality is always true (the statement \(-3 \geq -4\) is valid), the correct choice among options A, B, C, D, and E would be:

E: or

This signifies that all values of \(x\) satisfy the inequality.

For the number line representing the solution to the inequality \(x - 3 \geq -4 + x\), which is true for all \(x\), you can represent this as follows:

1. Draw a straight line across the indicated range of values from -12 to 12.
2. Place solid dots at each integer point along the line to indicate that all values are included (since the inequality is satisfied for all \(x\)).

The representation should look something like this, with dots at each integer:

Number Line: 
-12 • —————— • -10 • —————— • -8 • —————— • -6 • —————— • -4 • —————— • -2 • —————— • 0 • —————— • 2 • —————— • 4 • —————— • 6 • —————— • 8 • —————— • 10 • —————— • 12

Key Points:

• Use solid dots for each integer to indicate that values are included in the solution.
• You could also shade the entire number line to further emphasize that all values are valid solutions.

Yes, since the inequality \(x - 3 \geq -4 + x\) holds true for all real numbers, the number line would extend from \(-12\) to \(12\), and you can represent it visually in the following way:

1. Draw a line that starts at \(-12\) and ends at \(12\).
2. Use solid dots at each integer point (or shade the entire line) to indicate that all values in that range are part of the solution.

Here’s how you can visualize it:

Number Line:
-12 ● —————— -10 ● —————— -8 ● —————— -6 ● —————— -4 ● —————— -2 ● —————— 0 ● —————— 2 ● —————— 4 ● —————— 6 ● —————— 8 ● —————— 10 ● —————— 12

Alternatively, if you want to show that all numbers between \(-\infty\) and \(+\infty\) satisfy the inequality, you could represent the entire line without specific endpoints, as follows:

<----•──────────────•──────────────•──────────────•──────────────•──────────────•---->
        -12             -10            -8            -6            -4            -2            0             2             4             6            8             10            12

Either way, the concept remains: since every value \(x\) satisfies the inequality, you can represent this visually with dots or a solid line extending from \(-12\) to \(12\) (or even further).

Since the inequality we are dealing with is \(x - 3 \geq -4 + x\), which simplifies to \(-3 \geq -4\), this means all real numbers satisfy the inequality.

If you want to indicate points on the number line and you're using solid dots, it means that points are included in the solution. In this case, it does not actually matter if you are circling the dots or not, as you would generally mark all integer points with solid dots if you wanted to indicate inclusion.

If you're showing a specific range and want to depict included endpoints for a typical inequality (like \(x \leq 12\) or \(x \geq -12\)), you would typically use:

• Solid dots for included points (use solid dots).
• Open circles for excluded points (not applicable here since all numbers are included).

Summary:

• Use solid dots at each integer point (from -12 to 12) to indicate that all values along the number line are included in the solution.

So, your final representation would look like this with solid dots:

Number Line:
-12 • —————— -10 • —————— -8 • —————— -6 • —————— -4 • —————— -2 • —————— 0 • —————— 2 • —————— 4 • —————— 6 • —————— 8 • —————— 10 • —————— 12

No circles are necessary since all points are included in the solution set.