Solve for x*x* 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.

minus, 3, x, minus, 10, is greater than, 11, or, minus, 34, is less than or equal to, minus, 3, x, minus, 10

−3*x*−10>11or

−34≤−3*x*−10

We are given the compound inequality:

\[
-3x - 10 > 11 \quad \text{or} \quad -34 \leq -3x - 10
\]

### Solve the first inequality:
\[
-3x - 10 > 11
\]
Add 10 to both sides:
\[
-3x > 21
\]
Now divide by -3, and since we are dividing by a negative number, reverse the inequality:
\[
x < -7
\]

### Solve the second inequality:
\[
-34 \leq -3x - 10
\]
Add 10 to both sides:
\[
-24 \leq -3x
\]
Now divide by -3, again reversing the inequality:
\[
x \geq 8
\]

### Combined solution:
We have:
1. \( x < -7 \)
2. \( x \geq 8 \)

Since there is an "or" in the compound inequality, the solution is the union of these two intervals:
\[
x \in (-\infty, -7) \cup [8, \infty)
\]

### Graph on the number line:
We will mark the interval \( (-\infty, -7) \) with an open circle at \( x = -7 \) (not including -7), and \( [8, \infty) \) with a closed circle at \( x = 8 \) (including 8).

Let me know if you'd like to see the graph or have any questions about the solution.

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### Related Questions:
1. How do we solve inequalities with absolute values?
2. What happens when dividing or multiplying inequalities by a negative number?
3. How do you combine inequalities when there is an "and" condition instead of "or"?
4. How would this solution change if both inequalities had the same direction of inequality?
5. Can we always solve inequalities by graphing the solution?

**Tip:** When solving inequalities, remember that multiplying or dividing by a negative number reverses the direction of the inequality.

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.

-3*x - 10 > 11 or -34 ≤ -3*x - 10

This guide helps you solve the compound inequality -3x - 10 > 11 or -34 ≤ -3x - 10. Follow a step-by-step breakdown of algebraic operations, including reversing inequalities when dividing by negative numbers. Ideal for students in Grades 8-10.

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