Question

graph the solution to the inequality on a number line
-2 less than or equal too -.25(X-8) less than 1

Answer 1

To solve the compound inequality \(-2 \leq -0.25(X - 8) < 1\), we will break it into two parts and solve each part separately.

### Step 1: Solve the left part of the compound inequality

\[
-2 \leq -0.25(X - 8)
\]

First, we'll multiply each part of the inequality by \(-1\) and remember to reverse the inequality sign:

\[
2 \geq 0.25(X - 8)
\]

Now, multiply both sides by \(4\) to eliminate the fraction:

\[
8 \geq X - 8
\]

Now, add \(8\) to both sides:

\[
16 \geq X
\]

This can be rewritten as:

\[
X \leq 16
\]

### Step 2: Solve the right part of the compound inequality

\[
-0.25(X - 8) < 1
\]

Again, multiply both sides by \(-1\) (and reverse the inequality):

\[
0.25(X - 8) > -1
\]

Multiply both sides by \(4\):

\[
X - 8 > -4
\]

Now, add \(8\) to both sides:

\[
X > 4
\]

### Combined Solution

Now we can combine the results:

\[
4 < X \leq 16
\]

### Step 3: Graph on a Number Line

To graph this solution on a number line:

1. Draw a number line and mark the points \(4\) and \(16\).
2. Use an open circle at \(4\) (to indicate that \(4\) is not included in the solution).
3. Use a closed circle at \(16\) (to indicate that \(16\) is included in the solution).
4. Shade the region of the number line that lies between \(4\) and \(16\).

### Number Line Representation

```
<----|----|----|----|----|----|----|----|----|----|----|----|---->
0 2 4 6 8 10 12 14 16 18 20 22

(-------------------]
```

This graph shows that \(X\) is greater than \(4\) but less than or equal to \(16\).