Math Problem Statement

(x-1)(x+1)>0

Solution

To solve the inequality:

(x1)(x+1)>0(x - 1)(x + 1) > 0

This is a quadratic inequality. Follow these steps:

Step 1: Find the critical points

The critical points are the values of xx that make each factor equal to 0. Set each factor to 0 and solve:

  • x1=0x=1x - 1 = 0 \quad \Rightarrow \quad x = 1
  • x+1=0x=1x + 1 = 0 \quad \Rightarrow \quad x = -1

So, the critical points are x=1x = 1 and x=1x = -1.

Step 2: Test the intervals

These critical points divide the number line into three intervals:

  • (,1)(-\infty, -1)
  • (1,1)(-1, 1)
  • (1,)(1, \infty)

Now, test a value from each interval to see where the inequality holds.

  1. Interval (,1)(-\infty, -1): Choose x=2x = -2. (21)(2+1)=(3)(1)=33>0(True)(-2 - 1)(-2 + 1) = (-3)(-1) = 3 \quad \Rightarrow \quad 3 > 0 \quad (\text{True})

  2. Interval (1,1)(-1, 1): Choose x=0x = 0. (01)(0+1)=(1)(1)=11>0(False)(0 - 1)(0 + 1) = (-1)(1) = -1 \quad \Rightarrow \quad -1 > 0 \quad (\text{False})

  3. Interval (1,)(1, \infty): Choose x=2x = 2. (21)(2+1)=(1)(3)=33>0(True)(2 - 1)(2 + 1) = (1)(3) = 3 \quad \Rightarrow \quad 3 > 0 \quad (\text{True})

Step 3: Exclude the critical points

Since the inequality is strictly greater than 0 (i.e., >>, not \geq), the critical points x=1x = -1 and x=1x = 1 are not included in the solution.

Final Answer:

The solution is: x(,1)(1,)x \in (-\infty, -1) \cup (1, \infty)

This means xx can be any value less than 1-1 or greater than 11, but not equal to 1-1 or 11.

Would you like further clarification on any steps?

Here are 5 related questions for further practice:

  1. Solve ( (x - 2)(x + 2)

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Inequalities
Factoring

Formulas

(x - a)(x + b) > 0
Critical Points

Theorems

Sign Test for Inequalities
Factoring Method

Suitable Grade Level

Grades 9-10