Math Problem Statement

(x + 9)2(x + 7) > 0

Solution

To solve the inequality (x+9)2(x+7)>0(x + 9)^2(x + 7) > 0, follow these steps:

Step 1: Identify the critical points

We first look for the values of xx that make the expression equal to 0. The expression is equal to 0 when:

(x+9)2=0or(x+7)=0(x + 9)^2 = 0 \quad \text{or} \quad (x + 7) = 0

  1. From (x+9)2=0(x + 9)^2 = 0, we get x=9x = -9.
  2. From (x+7)=0(x + 7) = 0, we get x=7x = -7.

Thus, the critical points are x=9x = -9 and x=7x = -7.

Step 2: Analyze the sign of the expression in intervals

We will now divide the real number line into intervals based on these critical points and determine the sign of the expression (x+9)2(x+7)(x + 9)^2(x + 7) in each interval.

The intervals are:

  1. x<9x < -9
  2. 9<x<7-9 < x < -7
  3. x>7x > -7

Step 3: Test the sign in each interval

We test a point in each interval:

  • For x<9x < -9 (let x=10x = -10):
    (x+9)2(x+7)=(10+9)2(10+7)=1×(3)=3(x + 9)^2(x + 7) = (-10 + 9)^2(-10 + 7) = 1 \times (-3) = -3 So, the expression is negative in this interval.

  • For 9<x<7-9 < x < -7 (let x=8x = -8):
    (x+9)2(x+7)=(8+9)2(8+7)=12×(1)=1(x + 9)^2(x + 7) = (-8 + 9)^2(-8 + 7) = 1^2 \times (-1) = -1 So, the expression is negative in this interval.

  • For x>7x > -7 (let x=0x = 0):
    (x+9)2(x+7)=(0+9)2(0+7)=92×7=81×7=567(x + 9)^2(x + 7) = (0 + 9)^2(0 + 7) = 9^2 \times 7 = 81 \times 7 = 567 So, the expression is positive in this interval.

Step 4: Conclusion

From the sign analysis, we find that the expression (x+9)2(x+7)(x + 9)^2(x + 7) is:

  • Negative for x<9x < -9
  • Negative for 9<x<7-9 < x < -7
  • Positive for x>7x > -7

Since the inequality is (x+9)2(x+7)>0(x + 9)^2(x + 7) > 0, we want the expression to be positive. Therefore, the solution is:

x>7x > -7

Step 5: Exclude the critical points

At x=9x = -9, (x+9)2=0(x + 9)^2 = 0, and at x=7x = -7, (x+7)=0(x + 7) = 0. At these points, the expression equals 0, not greater than 0, so we exclude them from the solution.

Thus, the solution to the inequality is:

x>7x > -7

Final Answer:

x(7,)x \in (-7, \infty)

Would you like further clarification on any of the steps? Here are some related questions you might find interesting:

  1. What does the square term (x+9)2(x + 9)^2 imply for the solution?
  2. How do we solve quadratic inequalities like (x+9)2(x+7)<0(x + 9)^2(x + 7) < 0?
  3. What is the behavior of the function near its critical points?
  4. How can you graphically represent the solution to this inequality?
  5. What would happen if we had a cubic term instead of a quadratic one?

Tip: Always check the behavior of the expression at critical points, as they often represent boundaries where the sign of the expression changes!

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Math Problem Analysis

Mathematical Concepts

Inequalities
Quadratic Equations
Sign Analysis
Critical Points

Formulas

(x + 9)² = 0
(x + 7) = 0

Theorems

Sign analysis method
Behavior of quadratic functions near critical points

Suitable Grade Level

Grades 9-11