https://file.aiquickdraw.com/mathbot/file/1727493848906vmrl1oqz.jpg

It seems the image has been successfully uploaded. Let me extract the contents and help solve the inequalities.The text extracted from the image appears to contain some formatting issues. Based on what I could interpret, here are the inequalities:

1. \( 5^{3 - x} < \frac{1}{125} \)
2. \( 7 - e^{5t - 2} > 5 \)

Let me now solve them step by step.

---

### Part (a) \( 5^{3 - x} < \frac{1}{125} \):

1. Recall that \( \frac{1}{125} \) can be written as \( 5^{-3} \), so the inequality becomes:
   \[
   5^{3 - x} < 5^{-3}
   \]

2. Since the bases are the same, we can equate the exponents (remembering to reverse the inequality because \( 5^x \) is an increasing function):
   \[
   3 - x < -3
   \]

3. Solve for \( x \):
   \[
   -x < -6
   \]
   \[
   x > 6
   \]

### Part (b) \( 7 - e^{5t - 2} > 5 \):

1. Subtract 7 from both sides:
   \[
   -e^{5t - 2} > -2
   \]

2. Multiply both sides by -1 (remembering to reverse the inequality):
   \[
   e^{5t - 2} < 2
   \]

3. Take the natural logarithm of both sides to isolate \( t \):
   \[
   5t - 2 < \ln(2)
   \]

4. Solve for \( t \):
   \[
   5t < \ln(2) + 2
   \]
   \[
   t < \frac{\ln(2) + 2}{5}
   \]

---

Let me know if you need further clarification or more details on these solutions!

Solve the following inequalities: a) 5^(3-x) < 1/125, b) 7 - e^(5t-2) > 5

Learn how to solve the inequalities involving exponential and logarithmic functions: 5^(3-x) < 1/125 and 7 - e^(5t-2) > 5. This problem covers key concepts of exponential equations, logarithms, and inequalities, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation