Question

Solve the compound inequality. Other than empty set​, use interval notation to express the solution set and graph the solution set on a number line.
negative 4 less than or equals minus 4 x minus 12 less than 4
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Part 1

Answer 1

To solve the compound inequality \(-4 \leq -4x - 12 < 4\), we can break it into two parts and solve each part separately.

### Part 1: Solve \(-4 \leq -4x - 12\)

1. Add 12 to both sides:
\[
-4 + 12 \leq -4x
\]
\[
8 \leq -4x
\]

2. Divide by \(-4\) (remember to flip the inequality sign):
\[
\frac{8}{-4} \geq x
\]
\[
-2 \geq x
\]
or
\[
x \leq -2
\]

### Part 2: Solve \(-4x - 12 < 4\)

1. Add 12 to both sides:
\[
-4x < 4 + 12
\]
\[
-4x < 16
\]

2. Divide by \(-4\) (again, flip the inequality sign):
\[
x > \frac{16}{-4}
\]
\[
x > -4
\]

### Combine the results:
From the two inequalities we found:
1. \(x \leq -2\)
2. \(x > -4\)

Combining these gives:
\[
-4 < x \leq -2
\]

### Interval Notation:
The solution set in interval notation is:
\[
(-4, -2]
\]

### Graphing on a Number Line:

To graph this on a number line:

1. Draw a number line.
2. Mark the points \(-4\) and \(-2\).
3. Use an open circle at \(-4\) (since -4 is not included) and a closed circle at \(-2\) (since -2 is included).
4. Shade the region between these two points.

Here is a rough representation:

```
<---|----|----|----|----|----|----|----|---->
-5 -4 -3 -2 -1 0 1 2
( ]
```

Thus, the complete solution set is:
\[
\text{Interval: } (-4, -2] \quad \text{Graph: as shown above.}
\]