Math Problem Statement

Solution

To solve the inequality involving exponential expressions, let's first rewrite and analyze it step-by-step. The given inequality is:

(15)2x+11x>(15)3\left( \frac{1}{5} \right)^{\frac{2x + 1}{1 - x}} > \left( \frac{1}{5} \right)^{-3}

Step-by-Step Solution:

  1. Understand the Bases: Both sides of the inequality have the same base, 15\frac{1}{5}. Since 15\frac{1}{5} is less than 1, the inequality direction will flip when dealing with exponents.

  2. Rewrite the Inequality: Given that the base is less than 1, the inequality: (15)2x+11x>(15)3\left( \frac{1}{5} \right)^{\frac{2x + 1}{1 - x}} > \left( \frac{1}{5} \right)^{-3} can be simplified by comparing the exponents. The inequality flips because the base 15\frac{1}{5} is less than 1: 2x+11x<3\frac{2x + 1}{1 - x} < -3

  3. Solve the Exponent Inequality: 2x+11x<3\frac{2x + 1}{1 - x} < -3

    To solve this, we need to isolate xx. Let's handle it step-by-step: 2x+11x<3\frac{2x + 1}{1 - x} < -3

    Cross-multiply to get rid of the fraction: 2x+1<3(1x)2x + 1 < -3(1 - x)

    Distribute 3-3 on the right side: 2x+1<3+3x2x + 1 < -3 + 3x

    Isolate xx by subtracting 2x2x from both sides: 1<3+x1 < -3 + x

    Add 3 to both sides: 4<x4 < x

    Hence, the solution is: x>4x > 4

Conclusion:

The inequality (15)2x+11x>(15)3\left( \frac{1}{5} \right)^{\frac{2x + 1}{1 - x}} > \left( \frac{1}{5} \right)^{-3} is satisfied for x>4x > 4.

Would you like any further details or have any questions?

Related Questions:

  1. How does the base of an exponential expression affect the direction of an inequality?
  2. What is the process for solving inequalities involving fractions?
  3. How can you simplify inequalities with negative exponents?
  4. How do you handle inequalities when the base is between 0 and 1?
  5. Can you provide a similar inequality problem to solve?
  6. How would the inequality change if the base were greater than 1?
  7. What steps should be taken if the inequality had different bases?
  8. How can inequalities involving exponents be graphed?

Tip:

When solving inequalities involving exponential expressions, always pay attention to the base. If the base is less than 1, the inequality direction will flip when comparing the exponents.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponential Functions
Inequalities
Fractional Exponents

Formulas

-

Theorems

-

Suitable Grade Level

Grades 10-12