To address the problem presented in the image, I will first interpret the equations.

From the photo, we have two inequalities and an equation:

1. $2x - 3 \leq \pi \leq 8$
2. $x^2 - 3x - 27$
3. $x \geq 8$

Let's break these down step by step.

1. Solve the inequality $2x - 3 \leq \pi \leq 8$

We will solve the two parts of the inequality:

• $2x - 3 \leq \pi$
• $\pi \leq 8$

Solving $2x - 3 \leq \pi$:

$2x \leq \pi + 3$ $x \leq \frac{\pi + 3}{2}$

Solving $\pi \leq 8$:

This directly gives us $\pi \leq 8$, which is approximately $3.14159 \leq 8$, and this is always true.

Thus, the condition for $x$ is: $x \leq \frac{\pi + 3}{2}$

2. Solving $x^2 - 3x - 27 = 0$

This is a quadratic equation, and we will solve it using the quadratic formula: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-27)}}{2(1)}$ $x = \frac{3 \pm \sqrt{9 + 108}}{2}$ $x = \frac{3 \pm \sqrt{117}}{2}$ $x = \frac{3 \pm 10.82}{2}$ $x_1 = \frac{3 + 10.82}{2} = 6.91 \quad \text{and} \quad x_2 = \frac{3 - 10.82}{2} = -3.91$

3. Condition $x \geq 8$

The final condition is that $x \geq 8$.

Conclusion

Based on the conditions:

• $x \leq \frac{\pi + 3}{2}$ (approximately $x \leq 3.57$)
• From the quadratic solution, $x$ is approximately $6.91$ or $-3.91$, but we need $x \geq 8$.

Therefore, no solution exists that satisfies both the inequality and the condition $x \geq 8$ at the same time.

Would you like more details or clarification on any step?

Here are 5 related questions you might consider:

1. How do you handle quadratic inequalities?
2. What happens if you change the value of $\pi$ to a more general constant?
3. How do you solve inequalities involving multiple conditions?
4. Can inequalities be solved graphically for better visualization?
5. How does the quadratic formula handle complex solutions?

Tip: Always check for consistency between inequality conditions to ensure that a valid solution exists across all parts of the problem.