Problem 65 - Algebra 1

Question

You will look for a pattern. What happens to the values of \(\frac{x^{2}+6}{x+2},(x-2),\) and \(\frac{10}{x+2}\) as \(x\) increases?

Step-by-Step Solution

Answer

For the expression \(\frac{x^{2}+6}{x+2}\), the value increases with increasing \(x\); for the expression \(x-2\), the value directly increases with increasing \(x\); and for the expression \(\frac{10}{x+2}\), the value decreases with larger \(x\).

Step 1: Analyzing the first expression \(\frac{x^{2}+6}{x+2}\)

Observe that as \(x\) increases, the value of \(x^{2}\) will increase more significantly than \(x+2\). However, because this is a fraction, the result will be influenced by both the numerator and denominator. The value of \(\frac{x^{2}+6}{x+2}\) will vary with increasing \(x\) but the value of the expression will increase overall.

Step 2: Analyzing the second expression \(x-2\)

The expression \(x-2\) is linear. As \(x\) increases, the result of the binomial \(x-2\) directly increases as well. Since there are no additional variables or factors involved, this is a straightforward direct relationship.

Step 3: Analyzing the third expression \(\frac{10}{x+2}\)

In this case, as \(x\) increases, the denominator 'x+2' increases as well. But due to the nature of fractions, as the denominator gets larger, the overall value of the fraction decreases. Thus, the result of \(\frac{10}{x+2}\) will decrease as \(x\) increases.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In the rational expression \( \frac{x^{2}+6}{x+2} \), as \( x \) increases, the polynomial \( x^2 \) grows faster than \( x \), leading to a larger numerator when compared to the denominator. This means the overall value of the expression will largely increase with \( x \), even though the denominator is also increasing. Understanding how the numerator and denominator interact is crucial to predicting the behavior of a rational expression.

• The numerator and denominator are key to the expression's behavior.
• The expression can have variables raised to different powers, impacting growth rate.
• As \( x \to \infty \), rational expressions often follow the growth pattern of the highest power terms in the numerator and denominator.

Recognizing patterns in rational expressions helps in simplifying complex algebraic expressions and predicting their behavior.

Linear Equations

Linear equations describe a straight-line relationship between variables. The expression \( x-2 \) is a simple linear equation where the relationship between \( x \) and the result is direct and predictable. As \( x \) increases, the result of \( x-2 \) also increases. This behavior is uniform and easy to calculate since it involves simple addition or subtraction.

• The equation is of the form \( ax+b \).
• \( x \) has a power of 1, indicating a linear growth.
• The slope of the line represents the rate at which the expression changes; in this case, the slope is 1.

Linear equations help in analyzing straightforward algebraic relationships and are foundational in solving more complex algebraic problems.

Mathematical Analysis

Mathematical analysis involves a detailed examination of mathematical structures. Analyzing the expression \( \frac{10}{x+2} \) provides insight into how fractions behave as variables change. As \( x \) increases, the denominator \( x+2 \) becomes larger. In a divided expression, a larger denominator leads to a smaller overall value, indicating a diminishing trend as \( x \) increases. This analysis demonstrates the inverse relationship between the fraction’s value and its denominator.

• The inverse relationship is a critical concept in mathematical analysis.
• This helps in predicting how rates of decrease happen in rational functions.
• Understanding these relationships is key to solving complex calculus problems.

Through mathematical analysis, students learn to assess changes systematically and anticipate the effects of growth or shrinkage in algebraic expressions.