To solve the inequality $(x - 4)(x + 1) > 0$ and graph its solution on a number line, we can proceed step by step:

Step 1: Determine the critical points

The critical points are the values of $x$ that make the factors $x - 4$ or $x + 1$ equal to zero:

• $x - 4 = 0 \implies x = 4$
• $x + 1 = 0 \implies x = -1$

Thus, the critical points are $x = -1$ and $x = 4$.

Step 2: Divide the number line into intervals

The critical points divide the number line into three intervals:

1. $x < -1$
2. $-1 < x < 4$
3. $x > 4$

Step 3: Test the sign of the expression in each interval

We test the sign of $(x - 4)(x + 1)$ in each interval by picking a test point from each interval and substituting it into the inequality:

1. Interval $x < -1$: Choose $x = -2$: $(x - 4)(x + 1) = (-2 - 4)(-2 + 1) = (-6)(-1) = 6 > 0$ The product is positive.

2. Interval $-1 < x < 4$: Choose $x = 0$: $(x - 4)(x + 1) = (0 - 4)(0 + 1) = (-4)(1) = -4 < 0$ The product is negative.

3. Interval $x > 4$: Choose $x = 5$: $(x - 4)(x + 1) = (5 - 4)(5 + 1) = (1)(6) = 6 > 0$ The product is positive.

Step 4: Combine the results

The inequality $(x - 4)(x + 1) > 0$ is satisfied when the product is positive. From the test results, this happens in the intervals:

• $x < -1$
• $x > 4$

Step 5: Graph the solution

On a number line:

• Use open circles at $x = -1$ and $x = 4$ because the inequality is strict ($>$, not $\geq$).
• Shade the regions $x < -1$ and $x > 4$.

If you'd like, I can create the graph for you. Let me know!

Questions for further understanding:

1. How does this method change for $\leq 0$ or $\geq 0$?
2. What is the significance of critical points in inequalities?
3. Can you extend this solution to a three-factor inequality, e.g., $(x - 4)(x + 1)(x - 2) > 0$?
4. How would this change if one of the factors were squared, e.g., $(x - 4)^2(x + 1) > 0$?
5. How does interval testing work when the inequality involves a negative sign?

Tip:

Always pay attention to whether the inequality is strict ($>$) or includes equality ($\geq$). This determines whether the critical points are included in the solution.