Problem 44 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x+y>3} \\ {x+y>-2} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set to the system of inequalities is the region above the lines \(y = -x + 3\) and \(y = -x + 2\) on the coordinate plane.

Step 1: Graphing the inequalities individually

Each of these inequalities can be rewritten as \(y > -x+3\) and \(y > -x+2\) respectively. These are in the form \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept. The slope of the line is -1 for both and the y-intercepts are 3 and 2 respectively. We graph these lines with solid borders because the inequalities include a 'greater than' condition. Also, the region of solutions for each inequality is above the respective line because the y terms in our inequalities are 'greater than'.

Step 2: Finding the intersection of the solution regions

Upon drawing the lines and shading the solution regions (which lie above the respective lines), we notice an overlap of the shaded regions corresponding to the two inequalities. The overlapping region gives us a region on the coordinate plane which satisfies both inequalities. This area is the solution set to the system of inequalities.

Step 3: Describing the solution set

Our solution set is a region of the coordinate plane bound by the lines \(y = -x + 3\) and \(y = -x + 2\). It lies above each of these lines because of the 'greater than' condition in the inequalities. Therefore, the solution to the system of inequalities is in this intersected area of the two regions.

Key Concepts

Coordinate Plane

Imagine a vast blank space where we can represent numbers, specifically in pairs. This space is known as the coordinate plane, a crucial component in graphing systems of inequalities. Think of it as a map for mathematics, where every location has a specific address—a pair of numbers called coordinates. These coordinates are a combination of an 'x' (horizontal) value and a 'y' (vertical) value, separated by a comma and enclosed in parentheses like so: \( (x, y) \).

The plane is divided into four segments known as quadrants, and these quadrants help us determine the sign of the coordinates. Any point to the right of the origin (the center of the plane where \(x=0\) and \(y=0\)) will have a positive 'x' value, while points to the left will have a negative 'x' value. Similarly, points above the origin have positive 'y' values, and points below have negative 'y' values. When graphing inequalities, we plot not just single points, but entire regions on this coordinate plane.

Solution Set of Inequalities

Unlike equations, which often have a specific solution, inequalities have a solution set. This solution set includes all the possible pairs of 'x' and 'y' that make the inequality true. When we have a system of inequalities, we’re looking for a common solution set—a sort of 'meeting area' where all inequalities' conditions are satisfied.

Visually, this set is represented as a shaded region on the coordinate plane. Each inequality contributes its own region, and where these regions overlap, we find the solution set for the system. It's like throwing a net over the possible solutions, with each inequality pulling the net in different directions. Only the parts of the plane that are covered by all nets at once belong to the solution set. This is a very powerful way of understanding multiple conditions simultaneously and finding where they agree.

Inequality Graphing

The magic of understanding systems of inequalities comes to life in the process of inequality graphing. The first step is to transform each inequality into an equation—just for a moment—to graph the boundary line. For each inequality, if it includes a 'greater than' or 'less than' symbol without an equal to \( ( >, < ) \), we use a dashed line for this boundary to indicate that points on the line are not included in the solution set. For inequalities with 'greater than or equal to' or 'less than or equal to' \( (\geq, \leq) \), a solid line is used to include points on the line as part of the solution set.

Next, we choose a test point, often \( (0,0) \), the origin, unless it's on the line, to determine which side of the line we should shade. If the test point satisfies the inequality, the region including the origin is shaded; otherwise, we shade the opposite side. Once all inequalities are graphed and corresponding regions shaded, the overlapping area reveals the solution set of the system. This visual representation is intuitive: it allows students to 'see' the solution and fold complex algebraic concepts into a simple map to be explored.