Question

Solve and graph: -16 ≤ 3x - 4<2

Answer 1

To solve the compound inequality \(-16 \leq 3x - 4 < 2\), we can break it down into two parts and solve each part step-by-step.

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

**Inequality:** \(-16 \leq 3x - 4\)

1. Add 4 to both sides:
\[
-16 + 4 \leq 3x
\]
\[
-12 \leq 3x
\]

2. Divide by 3:
\[
-4 \leq x
\]
This can be rewritten as:
\[
x \geq -4
\]

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

**Inequality:** \(3x - 4 < 2\)

1. Add 4 to both sides:
\[
3x - 4 + 4 < 2 + 4
\]
\[
3x < 6
\]

2. Divide by 3:
\[
x < 2
\]

### Step 3: Combine the results

From both parts, we have:
\[
-4 \leq x < 2
\]

### Step 4: Write the solution in interval notation

The solution can be expressed in interval notation as:
\[
[-4, 2)
\]

### Step 5: Graph the solution

To graph the solution, we represent the interval on a number line:

- Place a solid dot at \(-4\) (since \(-4\) is included in the solution).
- Place an open dot at \(2\) (since \(2\) is not included in the solution).
- Shade the area between \(-4\) and \(2\).

Here is a rough representation of the graph:

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

●==================
-4 2
```
Remember, the solid dot at \(-4\) indicates that \(-4\) is included in the solution, while the open dot at \(2\) indicates that \(2\) is not included in the solution.