Question

Solve the inequality both algebraically and graphically. Give the solution in interval notation and draw it on a number line graph.
StartFraction x minus 3 Over 4 EndFraction
less thaneight thirds

Answer 1

To solve the inequality \(\frac{x - 3}{4} < \frac{8}{3}\), we can follow these steps.

### Algebraic Solution:

1. **Multiply both sides by 4** (to eliminate the fraction). Remember that since 4 is positive, the direction of the inequality remains unchanged:

\[
x - 3 < \frac{8}{3} \cdot 4
\]

Calculating \(\frac{8}{3} \cdot 4\):

\[
\frac{8 \cdot 4}{3} = \frac{32}{3}
\]

So, the inequality now becomes:

\[
x - 3 < \frac{32}{3}
\]

2. **Add 3 to both sides**:

\[
x < \frac{32}{3} + 3
\]

Convert 3 into thirds:

\[
3 = \frac{9}{3}
\]

Now the right-hand side becomes:

\[
\frac{32}{3} + \frac{9}{3} = \frac{32 + 9}{3} = \frac{41}{3}
\]

Thus, the solution in interval notation is:

\[
(-\infty, \frac{41}{3})
\]

### Graphical Solution:

To represent this solution graphically on a number line:

1. Draw a horizontal line to represent the number line.
2. Mark the point \(\frac{41}{3}\) (approximately 13.67) on the number line.
3. Since the inequality is strict (\(<\)), use an open circle at \(\frac{41}{3}\) to indicate that this value is not included in the solution set.
4. Shade the area to the left of \(\frac{41}{3}\) to indicate that all numbers less than \(\frac{41}{3}\) are included in the solution.

### Number Line Representation

Here's a simple representation:

```
<----(---●----------------)--->
-∞ 13.67 +∞
```

Where:
- The open circle \(\circ\) at \(\frac{41}{3}\) indicates that this point is not included.
- The line extending leftwards indicates all values less than \(\frac{41}{3}\).

**Final Answer**: The solution set in interval notation is \((-∞, \frac{41}{3})\).