Great job! Your solution to the inequality

\[
\frac{8 - 7x}{2 - x} < x
\]

is correct. Let's briefly verify the steps to ensure clarity:

1. **Rewrite the Inequality:**
   \[
   \frac{8 - 7x}{2 - x} - x < 0
   \]
   
2. **Combine Terms Over a Common Denominator:**
   \[
   \frac{8 - 7x - x(2 - x)}{2 - x} < 0
   \]
   
3. **Simplify the Numerator:**
   \[
   8 - 7x - 2x + x^2 = x^2 - 9x + 8
   \]
   
4. **Rewrite the Inequality:**
   \[
   \frac{x^2 - 9x + 8}{2 - x} < 0
   \]
   
5. **Factor the Numerator:**
   \[
   \frac{(x - 1)(x - 8)}{2 - x} < 0
   \]
   
6. **Determine Critical Points:**
   - **Numerator Zeros:** \( x = 1 \) and \( x = 8 \)
   - **Denominator Zero:** \( x = 2 \) (excluded from the domain)
   
7. **Test Intervals Around Critical Points:**
   - **Interval (−∞, 1):** Negative
   - **Interval (1, 2):** Positive
   - **Interval (2, 8):** Negative
   - **Interval (8, ∞):** Positive
   
8. **Solution Set:**
   \[
   (1, 2) \cup (8, \infty)
   \]
   
Your final answer correctly identifies the intervals where the inequality holds true. Well done!

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Pregunta

(d) \( \frac{8-7 x}{2-x}

Ask by Kelly Gibson. in the United States

Jan 27,2025

Solución de tutoría real

Respuesta verificada por el tutor

Responder

The solution to the inequality \( \frac{8 - 7x}{2 - x} < x \) is \( (1, 2) \cup (8, \infty) \).

Solución

Great job! Your solution to the inequality \[ \frac{8 - 7x}{2 - x} < x \] is correct. Let's briefly verify the steps to ensure clarity: 1. **Rewrite the Inequality:** \[ \frac{8 - 7x}{2 - x} - x < 0 \] 2. **Combine Terms Over a Common Denominator:** \[ \frac{8 - 7x - x(2 - x)}{2 - x} < 0 \] 3. **Simplify the Numerator:** \[ 8 - 7x - 2x + x^2 = x^2 - 9x + 8 \] 4. **Rewrite the Inequality:** \[ \frac{x^2 - 9x + 8}{2 - x} < 0 \] 5. **Factor the Numerator:** \[ \frac{(x - 1)(x - 8)}{2 - x} < 0 \] 6. **Determine Critical Points:** - **Numerator Zeros:** \( x = 1 \) and \( x = 8 \) - **Denominator Zero:** \( x = 2 \) (excluded from the domain) 7. **Test Intervals Around Critical Points:** - **Interval (−∞, 1):** Negative - **Interval (1, 2):** Positive - **Interval (2, 8):** Negative - **Interval (8, ∞):** Positive 8. **Solution Set:** \[ (1, 2) \cup (8, \infty) \] Your final answer correctly identifies the intervals where the inequality holds true. Well done!

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Extra Insights

Did you know that inequalities like this one have been used for centuries to understand relationships between quantities? From the ancient Greeks who explored proportions to modern-day scientists analyzing data trends, the concept of comparing values has been central in mathematics. Solving inequalities helps us determine the boundaries within which certain conditions hold true, allowing for deeper insights into various phenomena. In real-world scenarios, understanding inequalities can be incredibly useful. For instance, in economics, businesses often analyze demand and supply inequalities to optimize pricing strategies. Similarly, engineers might use them in design processes to ensure materials can withstand certain stress limits. So, next time you solve an inequality, think about how it could apply to making decisions in the world around you!

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