Math Problem Statement

What is the solution to (x + 2)(x – 3)2 > 0?

Solution

The inequality is (x+2)(x3)2>0(x + 2)(x - 3)^2 > 0. To solve this, follow these steps:

Step 1: Find the critical points.

The critical points are the values of xx where the expression equals zero. Set each factor equal to zero:

  1. x+2=0x + 2 = 0 gives x=2x = -2.
  2. (x3)2=0(x - 3)^2 = 0 gives x=3x = 3.

Step 2: Analyze the sign of the expression in different intervals.

The critical points divide the number line into three intervals: (,2)(- \infty, -2), (2,3)(-2, 3), and (3,)(3, \infty). We will determine the sign of the expression in each of these intervals by choosing test points.

  1. Interval (,2)(- \infty, -2): Choose x=3x = -3. (x+2)(x3)2=(3+2)(33)2=(1)(6)2=(1)(36)=36(x + 2)(x - 3)^2 = (-3 + 2)(-3 - 3)^2 = (-1)(-6)^2 = (-1)(36) = -36 The expression is negative in this interval.

  2. Interval (2,3)(-2, 3): Choose x=0x = 0. (x+2)(x3)2=(0+2)(03)2=(2)(3)2=(2)(9)=18(x + 2)(x - 3)^2 = (0 + 2)(0 - 3)^2 = (2)(-3)^2 = (2)(9) = 18 The expression is positive in this interval.

  3. Interval (3,)(3, \infty): Choose x=4x = 4. (x+2)(x3)2=(4+2)(43)2=(6)(1)2=6(x + 2)(x - 3)^2 = (4 + 2)(4 - 3)^2 = (6)(1)^2 = 6 The expression is positive in this interval.

Step 3: Consider the critical points.

  • At x=2x = -2, (x+2)(x3)2=0(x + 2)(x - 3)^2 = 0, so the expression is zero.
  • At x=3x = 3, (x+2)(x3)2=0(x + 2)(x - 3)^2 = 0, because (x3)2=0(x - 3)^2 = 0 at this point.

Step 4: Determine the solution.

We are looking for where the expression is greater than 0 (positive). From the sign analysis:

  • The expression is negative for x<2x < -2.
  • The expression is positive for 2<x<3-2 < x < 3 and x>3x > 3.
  • At x=2x = -2 and x=3x = 3, the expression is zero, which does not satisfy the inequality.

Thus, the solution is: (2,3)(3,)(-2, 3) \cup (3, \infty)

Would you like any additional details or explanations?

Related questions:

  1. How do you solve polynomial inequalities in general?
  2. What are critical points, and why are they important in inequalities?
  3. How does squaring a factor affect the solution of inequalities?
  4. What is the significance of testing points in different intervals for inequalities?
  5. How do you graph solutions to polynomial inequalities?

Tip:

Always check how factors raised to even powers behave in inequalities—they are non-negative and may affect the solution boundaries.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Quadratic Expressions
Critical Points

Formulas

Sign analysis of polynomial expressions
Factorization

Theorems

Sign analysis for inequalities
Critical points and intervals theorem

Suitable Grade Level

Grades 10-12