Solve for xx and graph the solution on the number line below.
minus, 26, is greater than, minus, 2, x, minus, 10, and, minus, 30, is less than, minus, 2, x, minus, 10
−26>−2x−10 and −30<−2x−10

We are given two inequalities:

1. \(-26 > -2x - 10\)
2. \(-30 < -2x - 10\)

Let's solve each inequality for \(x\).

### 1. Solving \(-26 > -2x - 10\):

\[
-26 > -2x - 10
\]

First, add 10 to both sides to isolate the term involving \(x\):

\[
-26 + 10 > -2x
\]

\[
-16 > -2x
\]

Next, divide both sides by \(-2\). Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips:

\[
\frac{-16}{-2} < \frac{-2x}{-2}
\]

\[
8 < x
\]

So, we have:

\[
x > 8
\]

### 2. Solving \(-30 < -2x - 10\):

\[
-30 < -2x - 10
\]

Add 10 to both sides:

\[
-30 + 10 < -2x
\]

\[
-20 < -2x
\]

Now, divide by \(-2\), flipping the inequality sign:

\[
\frac{-20}{-2} > \frac{-2x}{-2}
\]

\[
10 > x
\]

So, we have:

\[
x < 10
\]

### Combined Solution:

The solution to both inequalities is:

\[
8 < x < 10
\]

### Graphing the Solution on the Number Line:

To graph \(8 < x < 10\) on the number line:

- Draw a number line.
- Mark the points 8 and 10.
- Draw an open circle at 8 and 10 to indicate that these points are not included.
- Shade the region between 8 and 10.

Would you like a visual graph of the solution, or do you have any questions? 

### Further Questions:

1. What happens when you change the direction of the inequalities?
2. How would the solution change if we had \(8 \leq x < 10\)?
3. Can you solve and graph the solution of \(4x - 7 > 5x - 12\)?
4. How would the inequality change if we multiplied both sides by a positive number?
5. What is the significance of the open circles on the number line?
6. What if the inequalities had the same coefficient but different constants?
7. How does flipping the inequality affect the graph?
8. Can you provide an example where both inequalities would have the same solution?

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

Learn how to solve the inequalities -26 > -2x - 10 and -30 < -2x - 10 step-by-step with explanations and graph the solution on a number line. This problem involves solving linear inequalities and graphing on the number line, suitable for students in Grades 7-9.

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