Problem 22 - Algebra 2

Question

COLLEGE For Exercises 22 and \(23,\) use the following information. Rosa's professor says that the midterm exam will count for 40\(\%\) of each student's grade and the final exam will count for 60\(\% .\) A score of at least 90 is required for an A. The inequality \(0.4 x+0.6 y \geq 90\) represents this situation, where \(x\) is the midterm score and \(y\) is the final exam score. Graph this inequality.

Step-by-Step Solution

Answer

Graph the line \(y = \frac{-2}{3}x + 150\) and shade above.

Step 1: Understand the problem

We need to graph the inequality \(0.4x + 0.6y \geq 90\). This inequality shows the relationship between the midterm score \(x\) and the final exam score \(y\) required to attain an overall grade of at least 90.

Step 2: Rewrite the inequality

To make it easier to graph, rewrite the inequality in the form \(y \geq mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Rearrange to isolate \(y\): \[0.4x + 0.6y \geq 90\] becomes \[0.6y \geq -0.4x + 90\] and then \[y \geq \frac{-0.4}{0.6}x + \frac{90}{0.6}\].

Step 3: Simplify the inequality

Simplify the expression from the previous step:\[y \geq \frac{-2}{3}x + 150\]. Now, the inequality is ready to be graphed.

Step 4: Graph the boundary line

Graph the line \(y = \frac{-2}{3}x + 150\). Start with the y-intercept, \(b = 150\), which is the point (0, 150). Then use the slope \(-\frac{2}{3}\) to find another point. From (0, 150), move down 2 units and right 3 units to reach (3, 148). Draw a straight line through these points.

Step 5: Shade the solution region

Since the inequality is \(y \geq \frac{-2}{3}x + 150\), shade the region above the line. This shaded area represents all the combinations of midterm and final exam scores \((x, y)\) that result in an overall grade of 90 or more.

Key Concepts

Linear Equations

Linear equations are mathematical expressions where each term is either a constant or the product of a constant and a single variable. They can be written in the standard form as \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Linear equations describe straight lines when graphed on a coordinate plane. Understanding linear equations is crucial because they form the basis for more complex topics in algebra. In this specific problem, the inequality \( 0.4x + 0.6y \geq 90 \) is derived from a linear equation, which makes it essential to understand its components before graphing.
It is important to notice that linear equations reveal a constant rate of change, which appears as the slope of the line when graphed. In simple terms, the slope tells us how steep the line is and in which direction it moves.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( m \) shows the rate at which \( y \) changes concerning \( x \), while the y-intercept \( b \) indicates where the line crosses the y-axis.
Rewriting an inequality or equation in slope-intercept form makes it easier to graph and visualize. For instance, by transforming the original inequality to \( y \geq \frac{-2}{3}x + 150 \), we can effortlessly determine both the slope and the y-intercept for graphing. By observing the slope, \(-\frac{2}{3}\), we know that for each three units we move right on the x-axis, we'll move down two units on the y-axis.

• The slope shows how the two variables relate.
• The y-intercept provides a starting point for graphing.

These details are crucial for understanding how changes in one variable influence the other within a graph.

Inequality Solution

An inequality solution includes all the possible values that satisfy the inequality statement. Unlike equations, inequalities do not express precise equality but rather a range of possible solutions.
The inequality \( y \geq \frac{-2}{3}x + 150 \) encompasses all the pairs \((x, y)\) that make the expression true. Once we graph the boundary line \( y = \frac{-2}{3}x + 150 \), we must consider whether to shade the area above or below the line.
In this situation, the inequality is "greater than or equal to," so we shade above the line. This shading visually represents all midterm and final scores that would result in Rosa achieving the A grade.

Coordinate Plane

A coordinate plane is a two-dimensional plane where each point is identified with a pair of numerical coordinates \((x, y)\). These coordinates show the horizontal and vertical distances from the origin, which is the point \((0,0)\) where the x-axis and y-axis intersect.
The coordinate plane is pivotal in graphing equations and inequalities because it enables the visualization of mathematical relationships. For this exercise, the coordinate plane is used to plot the inequality \( y \geq \frac{-2}{3}x + 150 \).

• The x-axis represents the midterm exam score.
• The y-axis represents the final exam score.

Plotting and shading on this plane allows us to visually interpret the sets of scores that achieve an A grade, facilitating better understanding and analysis of the inequality's solution.