Question

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

Answer 1

To solve the compound inequality \(-11 \leq 16j + 7 - 7j < 16\), let's first simplify the middle expression \(16j + 7 - 7j\).

### Step 1: Simplify the inequality
Combine like terms:
\[
16j - 7j + 7 = 9j + 7
\]
Now our compound inequality reads:
\[
-11 \leq 9j + 7 < 16
\]

### Step 2: Break it into two parts
We can break this compound inequality into two separate inequalities:
1. \(-11 \leq 9j + 7\)
2. \(9j + 7 < 16\)

### Step 3: Solve the first inequality
\(-11 \leq 9j + 7\):
\[
-11 - 7 \leq 9j \quad \text{(subtract 7 from both sides)}
\]
\[
-18 \leq 9j
\]
\[
-2 \leq j \quad \text{(divide both sides by 9, remembering the inequality direction stays the same)}
\]

### Step 4: Solve the second inequality
\(9j + 7 < 16\):
\[
9j < 16 - 7 \quad \text{(subtract 7 from both sides)}
\]
\[
9j < 9
\]
\[
j < 1 \quad \text{(divide both sides by 9)}
\]

### Step 5: Combine the results
Putting the two solutions together, we have:
\[
-2 \leq j < 1
\]

### Step 6: Graph the solution
To graph the solution, we plot the inequality on a number line:

1. **Endpoints**:
- At \(j = -2\), we have a closed circle (since \(-2\) is included in the solution).
- At \(j = 1\), we have an open circle (since \(1\) is not included in the solution).

2. **Shade the region between** these values.

### Conclusion
The solution to the inequality is:
\[
[-2, 1)
\]

### Graph:
Here’s a representation of the solution on a number line:

```
<----●=================○---->
-2 1
```

This indicates all values of \(j\) between -2 and 1, including -2 but not including 1.