We are given two inequalities to solve:

1. $-2x - 10 \geq -22$
2. $-2x - 10 > -34$

Step 1: Solve the first inequality $-2x - 10 \geq -22$

Add 10 to both sides:

$-2x \geq -22 + 10$ $-2x \geq -12$

Now, divide both sides by $-2$. Remember to flip the inequality sign when dividing by a negative number:

$x \leq \frac{-12}{-2}$ $x \leq 6$

Step 2: Solve the second inequality $-2x - 10 > -34$

Add 10 to both sides:

$-2x > -34 + 10$ $-2x > -24$

Now, divide both sides by $-2$ and flip the inequality:

$x < \frac{-24}{-2}$ $x < 12$

Step 3: Combine the two inequalities

We now have:

1. $x \leq 6$
2. $x < 12$

To find the solution, we need the values of $x$ that satisfy both inequalities. The solution is the overlap between $x \leq 6$ and $x < 12$, which is:

$x \leq 6$

Step 4: Graph the solution on the number line

On the number line, the solution is all values less than or equal to 6. This is represented with a closed circle at 6 (since it includes 6) and shading to the left of 6.

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Would you like to explore any additional details or clarifications about this solution? Here are some related questions:

1. What happens to the inequality if we multiply or divide by a positive number instead of a negative number?
2. How do you graph inequalities that have strict inequalities ($<$ or $>$) versus non-strict inequalities ($\leq$ or $\geq$)?
3. How do we solve compound inequalities that involve "and" or "or"?
4. What is the difference between a solution set represented on a number line versus interval notation?
5. How do we check the solution of an inequality with a test point?

Tip: When solving inequalities, always pay special attention to the direction of the inequality sign when multiplying or dividing by a negative number.