Problem 98 - College Algebra with Modeling and Visualization

Question

The following data are exactly linear. $$ \begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 0.4 & 3.5 & 6.6 & 9.7 & 12.8 \end{array} $$ (a) Find a linear function \(f\) that models the data. (b) Solve the inequality \(2 \leq f(x) \leq 8\)

Step-by-Step Solution

Answer

The linear function is \(f(x) = 3.1x - 2.7\); the solution to the inequality is \(1.516 \leq x \leq 3.452\).

Step 1: Understanding Linear Function Form

A linear function is generally expressed as \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. To find these, we need to calculate \(m\) and \(b\) from the data.

Step 2: Calculate the Slope \(m\)

To find the slope \(m\), use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((1, 0.4)\) and \((2, 3.5)\), we compute: \[ m = \frac{3.5 - 0.4}{2 - 1} = 3.1 \] Thus, the slope \(m\) is 3.1.

Step 3: Calculate the Y-Intercept \(b\)

Choose a point, say \((1, 0.4)\), and use the formula \(y = mx + b\). Substitute \(m = 3.1\), \(x = 1\), and \(y = 0.4\): \[ 0.4 = 3.1 \cdot 1 + b \] Solving for \(b\), we get: \[ b = 0.4 - 3.1 = -2.7 \] The y-intercept \(b\) is -2.7.

Step 4: Formulate the Linear Function

Using the slope and y-intercept, the linear model is: \( f(x) = 3.1x - 2.7 \).

Step 5: Solve the Inequality

We need to solve the compound inequality \(2 \leq f(x) \leq 8\). Substitute \(f(x) = 3.1x - 2.7\) into the inequality:\[ 2 \leq 3.1x - 2.7 \leq 8 \] Step 1: Solve \(2 \leq 3.1x - 2.7\):\[ 3.1x \geq 4.7 \Rightarrow x \geq \frac{4.7}{3.1} \approx 1.516 \]Step 2: Solve \(3.1x - 2.7 \leq 8\):\[ 3.1x \leq 10.7 \Rightarrow x \leq \frac{10.7}{3.1} \approx 3.452 \]

Step 6: Write the Solution to the Inequality

The solution to the inequality is:\[ 1.516 \leq x \leq 3.452 \]This is the range of \(x\) values for which the inequality \(2 \leq f(x) \leq 8\) is true.

Key Concepts

Slope Calculation

When working with linear functions, the slope is a crucial factor. It tells us how steep the line is and the direction it slopes. In mathematical terms, the slope is represented as \(m\). For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope is calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here's a breakdown of how the calculation works:

• **Change in Y**: This is found by subtracting the first \(y\)-coordinate from the second: \(y_2 - y_1\).
• **Change in X**: Subtract the first \(x\)-coordinate from the second: \(x_2 - x_1\).
• Divide the change in \(y\) by the change in \(x\) to get the slope \(m\).

The slope reflects how \(y\) changes for every unit change in \(x\). In our data, calculating between two points \((1, 0.4)\) and \((2, 3.5)\), resulted in slope \(m = 3.1\). This means for every 1 unit increase in \(x\), \(y\) increases by 3.1 units.

Y-Intercept Determination

The y-intercept of a linear function is where the line crosses the \(y\)-axis. This is typically symbolized as \(b\) in the line's equation \(f(x) = mx + b\). Understanding and finding \(b\) helps you chart the whole line accurately without needing to plot multiple points.To find the y-intercept:

• **Choose an \((x,y)\) pair**: You can use any point from your data for this purpose. In our example, we used \((1, 0.4)\).
• **Substitute** the values into the linear equation after inserting the known slope \(m\).
• **Solve for \(b\)**: Rearrange the equation to find the value of \(b\).

From our example:

• Start with \(0.4 = 3.1 \cdot 1 + b\)
• Rearrange to solve for \(b\): \(b = 0.4 - 3.1 = -2.7\)

The calculated y-intercept \(-2.7\) tells us that the line will cross the \(y\)-axis at \(y = -2.7\). This point is crucial as it provides a starting point for plotting the entire line.

Solving Inequalities

Inequalities help us to determine a range of possible values for a variable, rather than a single solution. When dealing with a linear function like \(f(x) = 3.1x - 2.7\), we can use inequalities to find the range of \(x\) values that will satisfy certain conditions.Consider the inequality \(2 \leq f(x) \leq 8\):

• This is a compound inequality which means we need to solve it in two parts.
• Firstly, solve \(2 \leq 3.1x - 2.7\) to find the lower bound.
• Secondly, solve \(3.1x - 2.7 \leq 8\) for the upper bound.

For the lower bound:

• Add 2.7 to both sides: \(2 + 2.7 \leq 3.1x\)
• Simplify and divide by 3.1: \(x \geq \frac{4.7}{3.1} \approx 1.516\)

For the upper bound:

• Add 2.7 to both sides: \(3.1x \leq 8 + 2.7\)
• Simplify and divide by 3.1: \(x \leq \frac{10.7}{3.1} \approx 3.452\)

Thus, the acceptable range for \(x\) values is \(1.516 \leq x \leq 3.452\). This means only \(x\) values within this range will satisfy our original inequality.