Solve the inequality
(
x
-
2
)
(
x
-
3
)
2
<
0

Select all the intervals below that are included in the solution.
x>3
x<2
2<x<3
x<3
x>2

We are tasked with solving the inequality \((x - 2)(x - 3)^2 < 0\).

### Step 1: Analyze the expression
We can break down the inequality by considering the factors involved:

\[
(x - 2)(x - 3)^2 < 0
\]

- \( (x - 2) \) is a linear factor that changes sign at \( x = 2 \).
- \( (x - 3)^2 \) is a squared term, which means it is always non-negative (i.e., \( (x - 3)^2 \geq 0 \)) and equals 0 at \( x = 3 \).

Thus, the expression \( (x - 3)^2 \) is zero at \( x = 3 \), positive for \( x \neq 3 \), and never negative.

### Step 2: Key points of sign change
The critical points are \( x = 2 \) and \( x = 3 \), where the factors equal zero.

Now, divide the real line into intervals based on these points:

1. \( x < 2 \)
2. \( 2 < x < 3 \)
3. \( x > 3 \)

### Step 3: Determine signs of the product in each interval
We test the sign of \( (x - 2)(x - 3)^2 \) in each interval:

- **For \( x < 2 \):** Both \( x - 2 \) and \( (x - 3)^2 \) are positive, so the product is positive.
- **For \( 2 < x < 3 \):** Here, \( x - 2 \) is positive and \( (x - 3)^2 \) is positive (because it is always positive), so the product is negative.
- **For \( x > 3 \):** Both terms are positive, so the product is positive.

At \( x = 2 \), the expression equals zero, and at \( x = 3 \), the squared term forces the entire product to be zero as well.

### Step 4: Solution
The inequality is satisfied where the product is **negative**. This happens only in the interval \( 2 < x < 3 \).

Thus, the solution is \( 2 < x < 3 \).

### Step 5: Select the correct intervals
Now we check the given options against this solution:

- \( x > 3 \): Not included (product is positive).
- \( x < 2 \): Not included (product is positive).
- \( 2 < x < 3 \): **Included** (product is negative).
- \( x < 3 \): Not included (this would include the interval \( x < 2 \), where the product is positive).
- \( x > 2 \): Not included (includes the region \( x > 3 \), where the product is positive).

### Correct Interval:
The correct interval is:
- \( 2 < x < 3 \).

Would you like further details on any step or have any questions? 

Here are five related questions:
1. How do you solve inequalities involving absolute values?
2. How would the solution change if the inequality were \((x - 2)(x - 3)^2 \leq 0\)?
3. How can the discriminant help in solving quadratic inequalities?
4. How do you find the critical points in polynomial inequalities?
5. How does squaring a factor in an inequality affect the solution?

**Tip:** Always pay attention to squared terms in inequalities, as they never change sign except at their roots.

Solve the inequality (x - 2)(x - 3)^2 < 0. Select all the intervals below that are included in the solution. x>3, x<2, 2<x<3, x<3, x>2.

Learn how to solve the quadratic inequality (x - 2)(x - 3)^2 < 0 by analyzing critical points and testing intervals. Understand sign changes and the impact of squared terms. Suitable for students in Grades 9-12.

Math Problem Statement

Solve the inequality ( x

2 ) ( x

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation