solve and graph the inequality -4 less than or equal to x-5<-1

Let's solve the inequality:

\[
-4 \leq x - 5 < -1
\]

### Step 1: Solve the left side inequality \(-4 \leq x - 5\)
- Add 5 to both sides:

\[
-4 + 5 \leq x - 5 + 5
\]

\[
1 \leq x
\]

Thus, \(x \geq 1\).

### Step 2: Solve the right side inequality \(x - 5 < -1\)
- Add 5 to both sides:

\[
x - 5 + 5 < -1 + 5
\]

\[
x < 4
\]

### Step 3: Combine the inequalities
From both inequalities, we get:

\[
1 \leq x < 4
\]

So the solution is:

\[
x \in [1, 4)
\]

### Step 4: Graph the solution
- Draw a number line.
- Mark a solid point at \(x = 1\) to indicate that \(x = 1\) is included (since \( \leq \)).
- Mark an open point at \(x = 4\) to indicate that \(x = 4\) is not included (since \( < \)).
- Shade the region between 1 and 4.

Here is a description of the graph:

- The number line will show values from \(x = 1\) to \(x = 4\), with a solid circle at 1 and an open circle at 4.
- The region between the points is shaded, indicating the values \(x \in [1, 4)\).

Would you like a detailed graph visualization for this?

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Here are 5 related questions to expand on this:
1. How would the solution change if the inequality was strict, i.e., \(-4 < x - 5 < -1\)?
2. How do you solve compound inequalities involving three parts like this one?
3. Can you graph the inequality \(2x + 3 \leq 7 < 4x + 1\)?
4. How would you write the solution set in interval notation?
5. What are the key differences between solving equations and inequalities?

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

Discover how to solve and graph the compound inequality -4 ≤ x - 5 < -1 with a step-by-step explanation. This problem involves solving inequalities and using interval notation, suitable for students in Grades 6-8.

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