Problem 34 - Algebra 1

Question

In Exercises 34 and \(35,\) use the expression \(\frac{2 x-5}{x-2}\) and the table feature of a graphing calculator or spreadsheet software. Construct a table that shows the value of the numerator, the value of the denominator, and the value of the entire rational expression when the value of \(x\) is \(10,100,1000,10,000,100,000,\) and \(1,000,000\)

Step-by-Step Solution

Answer

After substituting each value of \(x\) into the equation, you'll have the desired table with the values of the numerator, denominator, and the entire expression for every value of \(x\).

Step 1: Substitute for the Numerator

Start with the given expression's numerator, which is \(2x-5\). Substitute each given \(x\) value, {10, 100, 1000, 10000, 100000, 1000000}, to this expression and record the results to form the numerator column of your table.

Step 2: Substitute for the Denominator

For the denominator of the given expression which is \(x-2\), again substitute each given \(x\) value, {10, 100, 1000, 10000, 100000, 1000000}, also record these values to form the denominator column of your table.

Step 3: Substitute for the Whole Expression

The whole expression is \(\frac{2x-5}{x-2}\). Substitute each value of \(x\) to the expression after finding the values of numerator and denominator. Divide the numerator by denominator for each corresponding \(x\) value. Record these results to form the 'value of entire expression' column in your table.

Key Concepts

Numerator and Denominator

Understanding rational expressions involves grasping the concepts of the numerator and the denominator. In the context of this exercise, the numerator is given by the linear expression \(2x - 5\), while the denominator is \(x - 2\).

• Numerator: This is the top part of a fraction. In the expression \(\frac{2x-5}{x-2}\), the numerator \(2x-5\) indicates that we perform operations of multiplication and subtraction to calculate its value for different \(x\) values. For example, if \(x = 10\), the numerator becomes \(2(10) - 5 = 15\).
• Denominator: This is the bottom part of a fraction. It's \(x-2\) in our expression. This means you subtract 2 from the \(x\) value to get the denominator. For \(x = 10\), the denominator yields \(10 - 2 = 8\).

Both of these components are crucial because they help us to evaluate the entire rational expression by dividing the numerator by the denominator across different \(x\) values.

Graphing Calculator

A graphing calculator can be a powerful tool for evaluating rational expressions, like the one provided in this exercise. It allows you to input expressions and automatically compute values for large sequences of numbers, which is convenient in constructing tables.Using the Table Function: Many graphing calculators have a table function where you can enter an expression and a sequence of \(x\) values. The calculator then outputs the corresponding values of the expression:

• Input: Enter the rational expression \(\frac{2x-5}{x-2}\) into the calculator.
• X-values: Enter \(10, 100, 1000, etc.\) as the \(x\) values.
• Output: The calculator will compute and display the numerator, denominator, and the rational expression’s values for these \(x\) values.

This saves time and reduces errors in calculation, making it easier to analyze trends and patterns.

Spreadsheet Software

Spreadsheet software, like Microsoft Excel or Google Sheets, is incredibly helpful for constructing tables and visualizing data involving rational expressions. You can automate calculations and see results in real-time as you change values.Steps to Utilize Spreadsheets:

• Set Up Columns: Create columns for \(x\) values, numerator, denominator, and the entire expression.
• Formula Input: In the numerator column, use a formula like =2*A1-5, assuming \(A\) holds your \(x\) values. For the denominator, use =A1-2.
• Rational Expression: To find \(\frac{2x-5}{x-2}\), input =B1/C1 in the appropriate column, where \(B\) and \(C\) are the numerator and denominator columns.

This not only helps in calculating but also allows you to easily manipulate and compare large data sets.

Substitution Method

The substitution method is a straightforward approach to evaluating expressions by replacing a variable with a given number. In this exercise, it is used to determine the values of the numerator, the denominator, and the entire expression for specific \(x\) values.Steps to Use Substitution:

• Numerator: Substitute each \(x\) value into \(2x-5\) to get the numerator values. For example, for \(x = 100\), substitute to get \(2(100) - 5 = 195\).
• Denominator: Similarly, for \(x=100\), substitute into \(x-2\) to get \(100-2=98\).
• Complete Expression: Finally, divide the numerator by the denominator to evaluate the full expression. For \(x=100\), the value is \(\frac{195}{98}\).

This systematic approach ensures consistency and accuracy across various \(x\) values, facilitating deeper understanding of how the expression behaves.