Problem 53 - Introductory Algebra for College Students

Question

The reason that systems of linear inequalities are appropriate for modeling healthy weight is because guidelines give healthy weight ranges, rather than specific weights, for various heights. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.

Step-by-Step Solution

Answer

The common solution region which satisfies both the inequalities \(y \geq x + 2\) and \(x \geq 1\) is the area to the right of the line \(x = 1\) and above the line \(y = x + 2\).

Step 1: Draw the Line for \(y = x + 2\)

Begin by drawing the line of \(y = x + 2\). It's a straight line with a slope of 1 and y-intercept at 2. The equality line is required only as a reference as the inequality we are interested in resides in the area either above or below this line.

Step 2: Determine Solution Region for \(y \geq x + 2\)

The given inequality is \(y \geq x + 2\). So, the solution will lie above the line \(y = x + 2\). This includes all the points on the line as well, because of the 'greater than or equal to' condition.

Step 3: Draw the Line for \(x = 1\)

Draw a vertical line passing through x = 1. It's the line of equation \(x = 1\). As with the first equation, we need to understand that the inequality will lie either to the left or right, or include the line itself depending on the sign of inequality.

Step 4: Determine Solution Region for \(x \geq 1\)

The given inequality is \(x \geq 1\). So all the solutions will be to the right of the line \(x = 1\). This includes the points on the line since it's 'greater than or equal to'.

Step 5: Find the Common Solution Region

Inspect the graph and identify the common region that satisfies both conditions. This region is where both the shaded regions intersect and represent the solutions to the system of inequalities.

Key Concepts

Graphing Inequalities

Understanding how to graph linear inequalities is crucial when working with systems that model real-world situations, such as determining healthy weight ranges compared to specific weights. Graphing an inequality like \(y \geq x+2\) involves first drawing the related line \(y = x + 2\). This line serves as a boundary for the inequality. Since the inequality is \(y \geq x+2\), you shade above the line to show that all points in this area are included in the solution set. The line itself is also included in the solution set, indicated by drawing a solid line rather than a dashed one.

Similarly, to graph \(x \geq 1\), a vertical line is drawn through \(x = 1\). All points to the right of this line, including the line itself, represent the solution set to the inequality. Graphing both inequalities on the same set of axes will allow us to visualize the region where the two solution sets overlap, providing the solutions to the system of inequalities.

Solution Set for Inequalities

The solution set for a system of linear inequalities is the set of all possible solutions that satisfy all the inequalities at once. In our exercise, after graphing the inequalities \(y \geq x+2\) and \(x \geq 1\), we look for the region where the shaded areas intersect. This common region represents all the coordinates \((x, y)\) that make both inequalities true. Visualizing the solution set graphically is a powerful tool as it allows the immediate identification of all possible solutions without testing individual points. When interpreting such graphs, always remember that every point in the shaded region is a solution, and any point outside this region does not satisfy the system of inequalities.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept of the line. The slope indicates the steepness of the line and the direction it tilts, while the y-intercept is the point where the line crosses the y-axis. In our example, the line \(y = x + 2\) is in slope-intercept form with a slope of 1 and a y-intercept of 2. This shows that the line goes up one unit for every unit it moves to the right, producing a 45-degree angle with the axes. Understanding the slope-intercept form allows for a quick and accurate graphing process. It is essential for creating a precise visual representation of the relationship described by the linear equation or inequality.