Question

Solve for s and graph the solution.
3s–7≥5 or
s+19
4
≤5

Answer 1

To solve the inequality \(3s - 7 \geq 5\) and the inequality \( \frac{s + 19}{4} \leq 5\), let's solve each part step by step.

### 1. Solve the first inequality:

\[
3s - 7 \geq 5
\]

Add 7 to both sides:

\[
3s \geq 5 + 7
\]

\[
3s \geq 12
\]

Now, divide both sides by 3:

\[
s \geq 4
\]

### 2. Solve the second inequality:

\[
\frac{s + 19}{4} \leq 5
\]

Multiply both sides by 4 (note that this doesn’t change the inequality because 4 is positive):

\[
s + 19 \leq 20
\]

Subtract 19 from both sides:

\[
s \leq 1
\]

### Combine the results:

We have two inequalities:

1. \(s \geq 4\)
2. \(s \leq 1\)

The solution to the compound inequality \(3s - 7 \geq 5 \text{ or } \frac{s + 19}{4} \leq 5\) is \(s \geq 4\) or \(s \leq 1\).

### Graphing the solution:

On a number line:

- Shade to the right starting at \(s = 4\) (indicating that \(s\) can take any value greater than or equal to 4).
- Shade to the left starting at \(s = 1\) (indicating that \(s\) can take any value less than or equal to 1).

### Final Solution:

The solution in interval notation is:

\[
(-\infty, 1] \cup [4, \infty)
\]

### Visual representation:

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

- The left portion represents \(s \leq 1\).
- The right portion represents \(s \geq 4\).