Question

Solve for s and graph the solution
3s-7>_5 or s+19/4<_5
plot the endpoints

Answer 1

To solve the compound inequality \( 3s - 7 \geq 5 \) or \( \frac{s + 19}{4} \leq 5 \), we'll solve each inequality separately.

### Step 1: Solve the first inequality

\[
3s - 7 \geq 5
\]

Add 7 to both sides:

\[
3s \geq 5 + 7
\]
\[
3s \geq 12
\]

Now, divide by 3:

\[
s \geq 4
\]

### Step 2: Solve the second inequality

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

Multiply both sides by 4:

\[
s + 19 \leq 20
\]

Subtract 19 from both sides:

\[
s \leq 1
\]

### Step 3: Combine the solutions

The solutions from the inequalities are:

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

Since the original statement uses "or", the solution set is:

\[
s \leq 1 \quad \text{or} \quad s \geq 4
\]

### Step 4: Graph the solution

To graph the solution on a number line:

1. Draw a number line.
2. Indicate the point \( 1 \) with a closed dot and shade to the left to represent \( s \leq 1 \).
3. Indicate the point \( 4 \) with a closed dot and shade to the right to represent \( s \geq 4 \).

The final graph looks like this:

```
<--|----|----|----|----|----|----|----|---->
-2 0 1 2 3 4 5 6
*--------------------------------------->
*-------> *
```

- The asterisk (*) at 1 means it's included (closed dot).
- The asterisk (*) at 4 also means it's included (closed dot).
- The shading to the left of 1 indicates all values less than or equal to 1.
- The shading to the right of 4 indicates all values greater than or equal to 4.

This represents the solution \( s \leq 1 \) or \( s \geq 4 \).