Problem 60 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x \geq 0} \\ {y \geq 0} \\ {2 x+y<4} \\ {2 x-3 y \leq 6} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set to the system of inequalities is the overlapped region from each of the individual inequalities in the first quadrant.

Step 1: Analyzing each Inequality

Start by understanding each inequality independent of each other: \n1. \(x \geq 0\) This inequality includes all points where x is greater than or equal to zero. This will be the area to the right of the y-axis including the y-axis itself. \n2. \(y \geq 0\) The inequality includes all points where y is greater than or equal to zero. This is the area above the x-axis, including the x-axis itself. \n3. \(2x + y < 4\) When written in slope-intercept form, \(y = -2x + 4\). This is a line that will partition the plane into two, the solution set being the region below the line (since y <). \n4. \(2x - 3y \leq 6\) When written in slope-intercept form, \(y = \frac{2}{3}x - 2\). This is a line that will also partition the plane into two, the solution being the region above the line (since y >). It includes the line itself because the original inequality included an 'equal to' symbol.

Step 2: Plotting each Inequality on the same plane

Plot each inequality on the same coordinate grid. The 'greater than or equal' inequalities (\(x \geq 0\), \(y \geq 0\)) result in shaded regions. For \(2x + y < 4\), the region present below the line \(y = -2x + 4\) is shaded and for \(2x - 3y \leq 6\) the region present above the line \(y = \frac{2}{3}x - 2\) is shaded.

Step 3: Identifying the overlapped region

The solution to the system of inequalities would be the region on the coordinate grid where all the shaded regions from the individual inequalities overlap. This shared shaded region is the solution set.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface with a horizontal axis (x-axis) and a vertical axis (y-axis). They intersect at a point called the origin. This plane is divided into four quadrants with positive and negative values for x and y coordinates, and it's essential for graphing systems of inequalities.

When graphing inequalities like in the given exercise, you use the coordinate plane to plot the lines or curves that represent each inequality. The x-axis typically represents the independent variable, while the y-axis represents the dependent variable. Each point on the plane has a pair of coordinates \(x, y\), which corresponds to its position relative to the origin.

In graphing inequalities, you essentially turn these equations into borders that divide the plane into regions. How we shade these regions determines the set of all possible solutions that satisfy the inequalities simultaneously.

Inequality Shading

Inequality shading is a visual method used to represent the solution set of an inequality on a coordinate plane. Each inequality divides the plane into two halves. The half that represents the solution set to the inequality is shaded.

For example, an inequality like \(2x + y < 4\) implies that the values of \(y\) must be less than a certain value, creating a 'boundary' given by the equality \(2x + y = 4\). This boundary is graphed as a line, and the region below this line is shaded, representing all the points that satisfy the inequality.

When multiple inequalities are graphed together, as in systems of inequalities, the area where the shading overlaps represents the solution to the system. The resulting shaded area must consider all the constraints posed by each inequality, so it's the common ground that satisfies all the conditions simultaneously.

Slope-Intercept Form

Slope-intercept form is an equation of a line in the format \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, or the point where the line crosses the y-axis. This form is incredibly useful for graphing since it directly gives you the slope and the y-intercept, which are instrumental for determining what the line looks like on a coordinate plane.

For instance, in the exercise, the inequality \(2x + y < 4\) can be transformed to match the slope-intercept form, becoming \(y = -2x + 4\). Here, \(m = -2\) shows the line falls two units vertically for every one unit it moves horizontally, and the y-intercept \(b = 4\) shows that the line crosses the y-axis at 4. Knowing the slope and y-intercept makes it simpler to draw the line and understand how it will divide the coordinate plane for shading in the solutions of inequalities.