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:
Step-by-Step Solution:
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Understand the Bases: Both sides of the inequality have the same base, . Since is less than 1, the inequality direction will flip when dealing with exponents.
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Rewrite the Inequality: Given that the base is less than 1, the inequality: can be simplified by comparing the exponents. The inequality flips because the base is less than 1:
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Solve the Exponent Inequality:
To solve this, we need to isolate . Let's handle it step-by-step:
Cross-multiply to get rid of the fraction:
Distribute on the right side:
Isolate by subtracting from both sides:
Add 3 to both sides:
Hence, the solution is:
Conclusion:
The inequality is satisfied for .
Would you like any further details or have any questions?
Related Questions:
- How does the base of an exponential expression affect the direction of an inequality?
- What is the process for solving inequalities involving fractions?
- How can you simplify inequalities with negative exponents?
- How do you handle inequalities when the base is between 0 and 1?
- Can you provide a similar inequality problem to solve?
- How would the inequality change if the base were greater than 1?
- What steps should be taken if the inequality had different bases?
- 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.
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Inequalities
Fractional Exponents
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12