The inequality provided in the image is:

$x^4 - 6x^3 + 8x^2 + 9x - 18 \geq 0$

Let's solve it step by step:

Step 1: Analyze the inequality

We are solving for the values of $x$ such that: $x^4 - 6x^3 + 8x^2 + 9x - 18 \geq 0$

Step 2: Factorize the polynomial

We'll look for roots by attempting synthetic division or testing possible rational roots using the Rational Root Theorem. This step will help in breaking down the polynomial into simpler factors.

I will compute the factors and solve for $x$.import sympy as sp

Define the variable and the polynomial

x = sp.symbols('x') polynomial = x4 - 6*x3 + 8x**2 + 9x - 18

Factorize the polynomial

factors = sp.factor(polynomial)

factors(x - 3)(x - 2)(x2 - x - 3)(x - 3)(x - 2)(x2 - x - 3)successThe polynomial factors as:

$(x - 3)(x - 2)(x^2 - x - 3) \geq 0$

Now, we solve this inequality by determining the critical points and analyzing the sign of the expression in the intervals between them.

Step 3: Find the critical points

The roots are:

• $x - 3 = 0 \Rightarrow x = 3$
• $x - 2 = 0 \Rightarrow x = 2$
• $x^2 - x - 3 = 0$, solve this quadratic equation for $x$.

Let me solve for the roots of $x^2 - x - 3 = 0$.# Solve the quadratic equation x^2 - x - 3 = 0 quadratic_roots = sp.solve(x**2 - x - 3, x)

quadratic_roots[1/2 - sqrt(13)/2, 1/2 + sqrt(13)/2][1/2 - sqrt(13)/2, 1/2 + sqrt(13)/2]successThe roots of the quadratic $x^2 - x - 3 = 0$ are:

$x = \frac{1}{2} - \frac{\sqrt{13}}{2} \quad \text{and} \quad x = \frac{1}{2} + \frac{\sqrt{13}}{2}$

These approximate to: $x \approx -1.30 \quad \text{and} \quad x \approx 2.80$

Thus, the critical points are: $x = -1.30, \, x = 2, \, x = 2.80, \, x = 3$

Step 4: Test the intervals

We divide the number line into intervals based on the critical points: $(-\infty, -1.30), \, (-1.30, 2), \, (2, 2.80), \, (2.80, 3), \, (3, \infty)$ We'll test the sign of the polynomial in each interval to determine where the inequality holds.

Let me compute the signs.# Define the intervals for testing test_intervals = [-2, 0, 2.5, 2.9, 4] # Points in each interval signs = [polynomial.subs(x, point) for point in test_intervals]

signs[60, -18, -0.187500000000000, -0.225900000000006, 18][60, -18, -0.187500000000000, -0.225900000000006, 18]successThe signs of the polynomial in the test intervals are as follows:

• At $x = -2$ (in $(-\infty, -1.30)$): Positive ($60 > 0$)
• At $x = 0$ (in $(-1.30, 2)$): Negative ($-18 < 0$)
• At $x = 2.5$ (in $(2, 2.80)$): Negative ($-0.1875 < 0$)
• At $x = 2.9$ (in $(2.80, 3)$): Negative ($-0.2259 < 0$)
• At $x = 4$ (in $(3, \infty)$): Positive ($18 > 0$)

Step 5: Determine where the inequality holds

The inequality $x^4 - 6x^3 + 8x^2 + 9x - 18 \geq 0$ holds in intervals where the polynomial is positive or zero: $x \in (-\infty, -1.30] \cup [3, \infty)$

Would you like a graphical representation or further clarification on any step?

Here are 5 follow-up questions related to the topic:

1. How can I check the critical points numerically?
2. What are the steps to graph this polynomial?
3. Why does the inequality change its sign at each root?
4. How is the Rational Root Theorem applied to this problem?
5. Can you explain how to approximate irrational roots?

Tip: Always verify your solutions by testing values in each interval or by graphing the polynomial.