To solve the inequality \( x^7 > 1 \), let's approach it step by step.

### Step 1: Rewrite the inequality
We can rewrite the inequality as:
\[
x^7 > 1
\]
which implies:
\[
x^7 - 1 > 0
\]
Factoring the left-hand side gives us:
\[
(x - 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) > 0
\]
This is a product of two terms. The inequality will hold where this product is positive.

### Step 2: Analyze critical points
The critical point comes from solving \(x - 1 = 0\), which gives \(x = 1\).

### Step 3: Analyze intervals
Now, we need to test the sign of the product in different intervals created by the critical point \(x = 1\).

1. **For \(x > 1\):**
   - Both \(x - 1 > 0\) and \(x^6 + x^5 + \dots + 1 > 0\).
   - Hence, the product is positive.

2. **For \(x = 1\):**
   - The expression is zero, so the inequality does not hold at \(x = 1\).

3. **For \(x < 1\):**
   - \(x - 1 < 0\), but \(x^6 + x^5 + \dots + 1 > 0\) (this sum is always positive for any real \(x\)).
   - Hence, the product is negative.

### Step 4: Solution
The inequality \(x^7 > 1\) holds for:
\[
x > 1
\]

### Final Answer:
\[
x > 1
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Solve the inequality \( x^5 > 1 \).
2. Solve the inequality \( x^3 > 8 \).
3. Solve \( 2^x > 4 \).
4. Solve \( x^4 < 1 \).
5. Find the solution to \( x^7 = 1 \).

**Tip:** When solving polynomial inequalities, always consider factoring and critical points to help break down the intervals where the inequality holds.

Explore how to solve the polynomial inequality x^7 > 1 using factoring techniques and critical points analysis. This step-by-step solution is ideal for students in Grades 10-12 and covers key concepts in algebra and polynomial inequalities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation