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Magazine Chapter 8: Problem 63
Explain why \(\frac{x+2}{x-3}>0\) and \((x+2)(x-3)>0\) have the same solutions.
Short Answer
Expert verified
Both inequalities have the solution intervals \((-\infty, -2)\) and \((3, \infty)\).
Step by step solution
01
Understanding the Original Inequality
We start with the inequality \( \frac{x+2}{x-3} > 0 \). This compares the fraction to zero, which means we need the fraction to be positive. This occurs when both the numerator \((x + 2)\) and the denominator \((x - 3)\) are either both positive or both negative.
02
Analyzing the Sign of the Numerator and Denominator
For \( \frac{x+2}{x-3} > 0 \), consider two cases: when both \(x+2\) and \(x-3\) are positive, and when both are negative. In these cases, the signs multiply to give a positive fraction.
03
Alternative Form of the Inequality
Consider the inequality \((x+2)(x-3) > 0 \). This form requires the product of the two linear expressions \((x + 2)\) and \((x - 3)\) to be positive, which also happens when both factors are either positive or negative.
04
Solving the Inequalities
Solving \((x + 2)(x - 3) > 0\) involves identifying intervals where the signs of \((x + 2)\) and \((x - 3)\) are the same. The solutions are intervals: \((-\infty, -2)\) and \((3, \infty)\).
05
Conclusion on Solution Equivalence
Both inequalities have the same solution sets. When \(x \in (-\infty, -2)\), both expressions \(x + 2\) and \(x - 3\) are negative, making both forms positive. When \(x \in (3, \infty)\), both are positive. Hence, they produce the same intervals or solution sets.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerator and Denominator Analysis
Analyzing the numerator and denominator of a fraction is crucial when solving inequalities like \( \frac{x+2}{x-3} > 0 \). For a fraction to be positive:
- The numerator \( x+2 \) and the denominator \( x-3 \) should be either both positive or both negative.
- If \( x+2 > 0 \), then \( x > -2 \).
- If \( x-3 > 0 \), then \( x > 3 \).
- \( x+2 < 0 \), which gives \( x < -2 \).
- \( x-3 < 0 \), leading to \( x < 3 \).
Sign Analysis
Sign analysis is a method used to determine the intervals where an expression holds true. In our inequality \( \frac{x+2}{x-3} > 0 \), sign analysis involves checking:
- When \( x+2 \) and \( x-3 \) are positive.
- When \( x+2 \) and \( x-3 \) are negative.
- If both are positive, their product is positive.
- If both are negative, their product is still positive since multiplying two negative numbers yields a positive result.
Linear Expressions
Linear expressions are algebraic expressions where variables are raised only to the first power. Understanding their properties is key to solving inequalities like \( \frac{x+2}{x-3} > 0 \) or \( (x+2)(x-3) > 0 \). A linear expression takes the form \( ax+b \). In the inequalities above:
- \( x+2 \) advances as a simple linear expression where \( a = 1 \) and \( b = 2 \).
- \( x-3 \) is also linear with \( a = 1 \) and \( b = -3 \).
Solution Intervals
Solution intervals are the specific ranges of \( x \) that make an inequality true. For \( \frac{x+2}{x-3} > 0 \), the solution intervals are determined by investigating where the expressions \( x+2 \) and \( x-3 \) have the same sign:
- For \( x < -2 \), both expressions \( x+2 \) and \( x-3 \) are negative.
- For \( x > 3 \), both \( x+2 \) and \( x-3 \) are positive.
