Solve for xx and graph the solution on the number line below. If possible, resolve your answer to a single inequality. In case of no solution (\varnothing∅), leave the number line blank.
6, is less than or equal to, minus, x, minus, 3, and, minus, 13, is less than or equal to, minus, x, minus, 3
6≤−x−3and
−13≤−x−3

We are given the system of inequalities:

1. \( 6 \leq -x - 3 \)
2. \( -13 \leq -x - 3 \)

Let's solve each inequality step by step.

### 1. Solve \( 6 \leq -x - 3 \)

Add 3 to both sides to eliminate the constant on the right side:

\[
6 + 3 \leq -x - 3 + 3
\]
\[
9 \leq -x
\]

Now, multiply both sides by \(-1\) to isolate \(x\), and remember to reverse the inequality sign when multiplying by a negative number:

\[
-9 \geq x \quad \text{or equivalently} \quad x \leq -9
\]

### 2. Solve \( -13 \leq -x - 3 \)

Similarly, add 3 to both sides:

\[
-13 + 3 \leq -x - 3 + 3
\]
\[
-10 \leq -x
\]

Multiply both sides by \(-1\) and reverse the inequality sign:

\[
10 \geq x \quad \text{or equivalently} \quad x \leq 10
\]

### Combine the two inequalities:

We now have the inequalities:

\[
x \leq -9 \quad \text{and} \quad x \leq 10
\]

The solution to this system is the more restrictive of the two, which is \(x \leq -9\).

### Graphing the solution on the number line:

Since the solution is \(x \leq -9\), you would shade the number line starting from \(-9\) and extend to the left (towards negative infinity). Place a solid dot at \(-9\) to indicate that \(-9\) is included in the solution.

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Here are 5 related questions:
1. What happens if we have a different constant on the right side of the inequalities?
2. How do you graph the solution when there is no solution (\(\varnothing\))?
3. What if the inequality was strict, i.e., \(6 < -x - 3\)?
4. How does multiplying or dividing by a negative number affect inequalities?
5. How do we interpret inequalities with absolute values?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number in inequalities.

Solve for x and graph the solution on the number line below. If possible, resolve your answer to a single inequality. In case of no solution (∅), leave the number line blank. 6 ≤ -x - 3 and -13 ≤ -x - 3

Learn how to solve the system of inequalities 6 ≤ -x - 3 and -13 ≤ -x - 3. This problem covers key algebraic concepts such as solving inequalities, reversing the inequality sign, and graphing solutions on a number line. Suitable for students in Grades 8-10.

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