Problem 59 - Algebra A Combined Function

Question

Explain why \(\frac{x+2}{x-3}>0\) and \((x+2)(x-3)>0\) have the same solution sets.

Step-by-Step Solution

Answer

Both inequalities imply the conditions \(x < -2\) or \(x > 3\).

Step 1: Understanding Inequality

Consider the inequality \(\frac{x+2}{x-3}>0\). This inequality states that the fraction \(\frac{x+2}{x-3}\) must be positive. A fraction is positive either when both the numerator and the denominator are positive or both are negative.

Step 2: Analyzing Numerator and Denominator

For \(\frac{x+2}{x-3}>0\) to hold: - Both \(x+2>0\) and \(x-3>0\), which implies \(x>-2\) and \(x>3\) so \(x>3\) (the stronger condition) OR - Both \(x+2<0\) and \(x-3<0\), implying \(x<-2\).

Step 3: Examining Product Inequality

Next, consider the inequality \((x+2)(x-3)>0\). A product is positive when both factors have the same sign: - Both \(x+2>0\) and \(x-3>0\), which occurs when \(x>3\) OR - Both \(x+2<0\) and \(x-3<0\), which occurs when \(x<-2\).

Step 4: Comparing Solutions

Both inequalities, \(\frac{x+2}{x-3}>0\) and \((x+2)(x-3)>0\), lead to the same conditions for \(x\): 1. \(x > 3\)2. \(x < -2\)Hence, the solution sets for both inequalities are \(x < -2\) or \(x > 3\).

Key Concepts

Inequality Solution Sets

An inequality solution set refers to the collection of all values that satisfy a given inequality.
For the inequality \(\frac{x+2}{x-3}>0\), the task is to identify values of \(x\) that make the expression positive.

• When the numerator and the denominator of the fraction have the same sign, the fraction is positive.
• Conversely, if they have opposite signs, the fraction is negative.

This requires splitting the problem into two scenarios: both numerator and denominator are positive, or both are negative. In solving these inequalities, the solution sets we find for both cases are \(x > 3\) and \(x < -2\), which readily satisfy the conditions for inequality solution sets.
Remember, analyzing these solution sets help determine the range of \(x\) values that can fulfill the inequality. This process of identifying solution sets is crucial for solving rational inequalities.

Numerical Analysis

Numerical analysis in the context of inequalities involves examining numerical values and their interactions with variables.
For example, understanding \(\frac{x+2}{x-3}>0\) demands numerical insight into both the fraction and the product, \((x+2)(x-3)\).

• A positive fraction or product requires both numerator and denominator, or factors, to share a consistent sign.
• This logical observation forms the basis upon which we analyze these inequalities.

To gain deeper insights, think about the number line:

• Visualize the critical points, \(x = -2\) and \(x = 3\), which stem from the component parts of the inequalities.
• Assess which segments of the number line satisfy the positive product or fraction requirement.

Such analysis helps us translate abstract algebraic expressions into concrete numerical solutions.

Algebraic Expressions

Algebraic expressions are fundamental components in solving inequalities.
They include variables, constants, and mathematical operations, which form the essence of inequalities like \(\frac{x+2}{x-3}>0\) and \((x+2)(x-3)>0\).

• Understanding how to manipulate these expressions is key to identifying solutions.
• One common approach is looking at the expressions both as a fraction and a product, simplifying them to understand the behavior of the inequality.

By breaking down the algebraic expression into components like \(x+2\) and \(x-3\):

• We ask when each component is positive or negative.
• This analysis informs which parts of the number line form the solution set for the inequality.

Mastery of algebraic expressions allows for effective problem-solving, aiding in tackling complex inequalities by breaking them down step by step.