https://file.aiquickdraw.com/mathbot/file/1725470120926wddhbdof.jpg

Let's solve each part of the problem in the image.

### Part (a): Solve the Compound Inequality

The inequality given is:
\[
2 + 3x < 5x + 1 \leq 16
\]

This is a compound inequality and can be solved in two steps.

1. **Solve the left side of the inequality:**
   \[
   2 + 3x < 5x + 1
   \]
   Subtract \(3x\) from both sides:
   \[
   2 < 2x + 1
   \]
   Subtract 1 from both sides:
   \[
   1 < 2x
   \]
   Divide by 2:
   \[
   \frac{1}{2} < x
   \]
   So, \( x > \frac{1}{2} \).

2. **Solve the right side of the inequality:**
   \[
   5x + 1 \leq 16
   \]
   Subtract 1 from both sides:
   \[
   5x \leq 15
   \]
   Divide by 5:
   \[
   x \leq 3
   \]

So, the solution for part (a) is:
\[
\frac{1}{2} < x \leq 3
\]
or in interval notation:
\[
\left(\frac{1}{2}, 3\right]
\]

### Part (b): Solve the Inequality \( x^3 - x^2 - x + 1 > 0 \)

To solve the inequality \( x^3 - x^2 - x + 1 > 0 \), we first find the roots of the equation \( x^3 - x^2 - x + 1 = 0 \) by factoring or using numerical methods.

Let's factor the cubic expression:
\[
x^3 - x^2 - x + 1 = (x - 1)(x^2 + 1)
\]

Now, set \( x^3 - x^2 - x + 1 = 0 \):
\[
(x - 1)(x^2 + 1) = 0
\]

This gives us the roots:
- \( x = 1 \) (from \( x - 1 = 0 \))
- \( x^2 + 1 = 0 \) has no real solutions since \( x^2 = -1 \) is not possible for real \( x \).

The only real root is \( x = 1 \).

Now, analyze the sign of the expression around \( x = 1 \):

- For \( x < 1 \), \( (x - 1) < 0 \) and \( x^2 + 1 > 0 \), so the product is negative.
- For \( x > 1 \), \( (x - 1) > 0 \) and \( x^2 + 1 > 0 \), so the product is positive.

Therefore, the solution to the inequality is:
\[
x > 1
\]
or in interval notation:
\[
(1, \infty)
\]

Would you like further details on any of these solutions or have any questions?

Here are 5 related questions that expand on the information:

1. How would the solution to part (a) change if the inequality was \(2 + 3x \leq 5x + 1 < 16\)?
2. How can you verify the solution for the cubic inequality in part (b) graphically?
3. What is the significance of the root \(x = 1\) in the context of part (b)'s inequality?
4. Can the inequality in part (b) have any complex roots, and how do they affect the solution?
5. How would you solve a similar cubic inequality if it had more than one real root?

**Tip:** Always check the intervals around critical points (roots) when solving polynomial inequalities to determine the sign of the expression in each interval.

Solve the compound inequality 2 + 3x < 5x + 1 ≤ 16 and the cubic inequality x^3 - x^2 - x + 1 > 0.

Learn how to solve the compound inequality 2 + 3x < 5x + 1 ≤ 16 and the cubic inequality x^3 - x^2 - x + 1 > 0. This problem covers key algebraic concepts including linear inequalities and factoring cubic polynomials. Suitable for students in Grades 10-12.

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