Problem 97 - College Algebra with Modeling and Visualization

Question

The following data are exactly linear. $$ \begin{array}{cccc} x & 0 & 2 & 4 & 6 \\ y & -1.5 & 4.5 & 10.5 & 16.5 \end{array} $$ (a) Find a linear function \(f\) that models the data. (b) Solve the inequality \(f(x)>2.25\)

Step-by-Step Solution

Answer

The linear function is \(f(x) = 3x - 1.5\), and \( x > 1.25 \) for \(f(x) > 2.25\).

Step 1: Identify two points from the data

Choose two points from the given data table. For instance, we can take the first two points: \((0, -1.5)\) and \((2, 4.5)\).

Step 2: Calculate the slope (m) of the line

The slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4.5 - (-1.5)}{2 - 0} = \frac{6}{2} = 3 \]

Step 3: Determine the y-intercept (b)

To find the y-intercept \(b\), use the linear equation format \(y = mx + b\) with one of the points, like \((0, -1.5)\). Since \(x = 0\) at this point, we have \(b = -1.5\).

Step 4: Write the linear function f(x)

Now that we have both the slope and the y-intercept, we can write the equation of the line as: \[ f(x) = 3x - 1.5 \]

Step 5: Solve the inequality \(f(x) > 2.25\)

Substitute \(f(x) = 3x - 1.5\) into the inequality: \[ 3x - 1.5 > 2.25 \]Add 1.5 to both sides: \[ 3x > 3.75 \]Then divide by 3: \[ x > 1.25 \]

Key Concepts

Slope Calculation

Understanding slope is crucial because it tells us how steep a line is. When finding the slope of a line between two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula shows the change in the y-values divided by the change in the x-values, often referred to as "rise over run."For our exercise, we chose the points \((0, -1.5)\) and \((2, 4.5)\). Plugging these into the formula, we calculated the slope as:\[ m = \frac{4.5 - (-1.5)}{2 - 0} = \frac{6}{2} = 3 \]This tells us that for every unit of x increase, the y value increases by 3.
Knowing how to calculate slope is key to writing equations of lines and understanding changes in values within contexts like physics, economics, and various other fields.

Inequality Solving

Solving inequalities involves finding the range of values that satisfy a condition. In this exercise, we needed to solve the inequality: \[ f(x) > 2.25 \]where \( f(x) = 3x - 1.5 \).The first step is to substitute the linear function into the inequality, giving us: \[ 3x - 1.5 > 2.25 \]Then, we add 1.5 to both sides to isolate the term with \(x\):\[ 3x > 3.75 \]Finally, dividing each side by 3, we find:\[ x > 1.25 \]This means any \(x\) value greater than 1.25 will satisfy the inequality.
Understanding how to solve inequalities helps in determining such boundaries and can be useful in various real-world scenarios like maximum capacity calculations or financial constraints.

Graphing Linear Equations

Graphing linear equations involves representing them visually on a coordinate plane. The equation we derived is \[ f(x) = 3x - 1.5 \]This is a basic linear equation in the slope-intercept form \( y = mx + b \):- The slope \(m\) is 3, indicating the line rises 3 units up for every 1 unit moved to the right.- The y-intercept \(b\) is -1.5, showing where the line crosses the y-axis.To graph this:

• Start at the y-intercept (0, -1.5) on the graph.
• From this point, use the slope to find another point. Move up 3 units and 1 unit to the right, reaching (1, 1.5).
• Draw a line through these points, extending it on both ends.

Graphing linear functions can help visually analyze relationships and predict outcomes. It's a core skill in expressing mathematical relationships in statistics and various scientific fields.