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Questions and Answers
For an exponential function where 𝑏 > 1, if 𝑥1 > 𝑥2, what can be said about 𝑏?
For an exponential function where 𝑏 > 1, if 𝑥1 > 𝑥2, what can be said about 𝑏?
What is the key to solving exponential inequalities?
What is the key to solving exponential inequalities?
What is the solution set for the inequality 2𝑥+9 ≤ 4𝑥+1?
What is the solution set for the inequality 2𝑥+9 ≤ 4𝑥+1?
What is the solution set for the inequality (0.5)1/343 𝑥−5 ≤ 3(𝑥+8)?
What is the solution set for the inequality (0.5)1/343 𝑥−5 ≤ 3(𝑥+8)?
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What is the solution set for the inequality 3𝑥 > (0.25) −𝑥−2?
What is the solution set for the inequality 3𝑥 > (0.25) −𝑥−2?
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What is the solution set for the inequality 125 / (2𝑥+8) ≤ 512 / (𝑥+5)?
What is the solution set for the inequality 125 / (2𝑥+8) ≤ 512 / (𝑥+5)?
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Study Notes
Exponential Function Properties
- For an exponential function with base b greater than 1, if x1 is greater than x2, then b^x1 will be greater than b^x2.
Solving Exponential Inequalities
- The key to solving exponential inequalities lies in understanding the behavior of exponential functions with respect to their base.
- If the base is greater than 1, the function increases as the exponent increases.
- If the base is between 0 and 1, the function decreases as the exponent increases.
Solving Linear Inequalities
- The solution set for the inequality 2x + 9 ≤ 4x + 1 is x ≥ 4.
- To solve, isolate x on one side of the inequality.
Solving Exponential Inequalities with Fractional Bases
- The solution set for the inequality (0.5)^1/343 (x - 5) ≤ 3(x + 8) is x ≥ -2.94.
- Simplify the fractional base by converting it to a power of 1/2.
- Isolate x on one side of the inequality and remember to flip the inequality sign if multiplying or dividing both sides by a negative number.
Solving Exponential Inequalities with Fractional Exponents
- The solution set for the inequality 3x > (0.25)^-x-2 is x > -2.
- Rewrite 0.25 as 1/4 and express the power as a positive exponent.
- Isolate x on one side of the inequality and remember to flip the inequality sign if multiplying or dividing both sides by a negative number.
Solving Rational Inequalities
- The solution set for the inequality 125 / (2x + 8) ≤ 512 / (x + 5) is x < -5 or -5<x<11/2.
- Subtract the right side of the inequality from the left side and simplify to a single fraction.
- Find the values of x that make the numerator or denominator equal to zero.
- Test each interval to determine if the inequality holds true.
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Description
Test your understanding of solving exponential inequalities with this quiz. Practice solving inequalities involving exponential functions and different bases.
