Question

Solve for j and graph the solution.
–
11≤16j+7–7j<16

Answer 1

To solve the compound inequality \(-11 \leq 16j + 7 - 7j < 16\), we will first simplify the expressions within the inequalities.

### Step 1: Simplify the inequality
Combine the terms involving \(j\):
\[
16j - 7j = 9j
\]
Thus, the inequality becomes:
\[
-11 \leq 9j + 7 < 16
\]

### Step 2: Separate the inequalities
We will solve the left side and the right side separately:
1. **Left Side**:
\[
-11 \leq 9j + 7
\]
Subtract 7 from both sides:
\[
-11 - 7 \leq 9j
\]
Simplifying gives:
\[
-18 \leq 9j
\]
Now divide by 9:
\[
-2 \leq j \quad \text{or} \quad j \geq -2
\]

2. **Right Side**:
\[
9j + 7 < 16
\]
Subtract 7 from both sides:
\[
9j < 16 - 7
\]
This simplifies to:
\[
9j < 9
\]
Dividing by 9 gives:
\[
j < 1
\]

### Step 3: Combine the results
From our findings, we have:
\[
-2 \leq j < 1
\]
In interval notation, this is expressed as:
\[
[-2, 1)
\]

### Step 4: Graph the solution
To graph the solution on a number line:

1. Draw a horizontal line representing the real number line.
2. Mark the points \(-2\) and \(1\).
3. Since \(-2\) is included in the solution, you can represent it with a closed circle at \(-2\).
4. Since \(1\) is not included, represent it with an open circle at \(1\).
5. Shade the region between \(-2\) and \(1\).

Here’s a simple representation:

```
<---|---|---|---|---|---|---|---|---|---|---|--->
-3 -2 -1 0 1 2 3
●====)
```

The closed circle at \(-2\) indicates that \(-2\) is included, while the open circle at \(1\) indicates that \(1\) is not included in the solution set.