Problem 2 - Algebra 2 and Trigonometry

Question

Shelley said that if \((x-7)(x-5)<0,\) then \((x-7)\) must be the negative factor and \((x-5)\) must be the positive factor. a. Do you agree with Shelly? Explain why or why not. b. When the product of two factors is negative, is it always possible to tell which is the positive factor and which is the negative factor? Justify your answer.

Step-by-Step Solution

Answer

a. Yes, Shelly is right for interval \((5, 7)\). b. No, without specifics, the positive or negative factor cannot be definitively identified.

Step 1: Understand the Problem

Shelley's claim is based on the inequality \((x-7)(x-5)<0\). Shelly believes that one specific factor, \((x-7)\), is negative, and the other, \((x-5)\), is positive, leading to a negative product. We need to validate this reasoning.

Step 2: Identify the Zero Points

To understand when the product changes signs, find the points where each factor equals zero: For \((x-7)=0\), \(x=7\), and for \((x-5)=0\), \(x=5\). These points partition the number line into intervals.

Step 3: Test Intervals Around Zero Points

Test an example value from each interval to determine where the inequality holds:1. **Interval:** \(x < 5\) - Choose \(x = 4\): \((4-7)(4-5) = (-3)(-1) = 3 > 0\).2. **Interval:** \(5 < x < 7\) - Choose \(x = 6\): \((6-7)(6-5) = (-1)(1) = -1 < 0\).3. **Interval:** \(x > 7\) - Choose \(x = 8\): \((8-7)(8-5) = (1)(3) = 3 > 0\).The product is negative in the interval \((5, 7)\).

Step 4: Analyze the Results

In the negative interval \((5, 7)\), we noted that - \((x-7)<0\) (factor \(x-7\) is negative)- \((x-5)>0\) (factor \(x-5\) is positive)This supports Shelley's claim that in this interval, one factor is negative, the other positive.

Step 5: Generalize Conditions for Zero Products

Examine if Shelley's reasoning applies universally when the product of two factors is negative. In general, when \(ab < 0\), either \(a\) is negative while \(b\) is positive, or \(a\) is positive while \(b\) is negative. Without extra context, identifying which specific factor is positive or negative solely from \(ab < 0\) is not always possible unless further analyzed using factor specifics as done in Step 3.

Key Concepts

Factorization

Factorization is the process of breaking down an expression into a product of simpler expressions called factors. In algebra, particularly when solving inequalities like \((x-7)(x-5)<0\), factorization helps us identify key components that determine the behavior of the expression across different intervals of interest.

To factorize correctly, recognize patterns or common elements in the expression. For the given inequality, \((x-7)(x-5),\) both terms are linear factors, meaning they each contain a variable raised to the first power. This structure allows us to dissect the inequality into two distinct elements, "\((x-7)\)" and "\((x-5)\)," which we'll individually examine. Each factor independently contributes to the sign of the overall product, allowing us to find solutions by testing changes in sign.

Knowing how to factorize is crucial because it forms the foundation for discovering zero points, which further guide us in analyzing the problem efficiently.

Zero Points

Zero points, also known as roots or x-intercepts, are crucial when solving mathematical inequalities. These are specific values of x that make each factor of an inequality equal to zero. Identifying zero points is the first step in interval testing.

For the inequality \((x-7)(x-5)<0,\) the zero points are calculated by setting each factor to zero :

• For \((x-7)=0,\) the zero point is \(x=7.\)
• For \((x-5)=0,\) the zero point is \(x=5.\)

These zero points divide the x-axis into distinct intervals, providing a framework to analyze the behavior of the inequality. Understanding zero points allows you to determine where the factors change signs, which is critical in identifying where the inequality holds true.

Interval Testing

Interval testing comes after identifying zero points and is an essential method to determine the sign of the product in each interval. Once the zero points are known, the number line is divided into segments defined by these points. Each interval can potentially lead to different behaviors of the inequality, thus needs separate testing.

Given the intervals from the zero points \(x = 5\) and \(x = 7\):

• Interval: \(x < 5\), such as \(x = 4\).
• Interval: \(5 < x < 7\), such as \(x = 6\).
• Interval: \(x > 7\), such as \(x = 8\).

By choosing a test point within each interval, assess the signs of \((x-7)\) and \((x-5)\), and subsequently the sign of their product. Interval testing simplifies understanding which intervals satisfy the inequality.

Sign Analysis

Sign analysis refers to examining how the signs of terms can lead to a positive or negative product in a mathematical expression. It is particularly useful when dealing with inequalities and factorization.

When facing an inequality like \((x-7)(x-5)<0,\) perform sign analysis for each identified interval to confirm where the inequality holds. During this process, check the sign of each factor and apply this to understand their combined effect:

• In interval \((x < 5),\) both factors are negative, so their product is positive.
• In interval \((5 < x < 7),\) one factor is negative and the other positive, leading to a negative product.
• In interval \((x > 7),\) both factors are positive, so their product is again positive.

Through sign analysis, you systematically conclude that the inequality is only satisfied in the interval from 5 to 7, confirming Shelley's observations within this restricted range.