Problem 59 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+5 y<10 \\ 3 x+4 y \leq 12 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the overlapped region in the first quadrant enclosed by the lines \(2x+5y=10\) and \(3x+4y=12\), including the line \(3x+4y=12\) but excluding the line \(2x+5y=10\).

Step 1: Analyzing inequalities

This step involves understanding each of the given inequalities. The first two, \(x \geq 0\) and \(y \geq 0\), represent the first quadrant of the Cartesian plane where both x and y are non-negative. The third inequality, \(2x+5y<10\), represents all points that are below the line \(2x+5y=10\). Finally, the fourth inequality \(3x+4y\leq12\) means all the points are either on the line \(3x+4y=12\) or below it.

Step 2: Graphing the inequalities

Sketch all the inequalities on the same graph. Remember, excluding the boundary line for the inequality \(2x+5y<10\) and including the boundary line for the inequality \(3x+4y\leq12\). The solution will only be in the first quadrant because of the first two inequalities \(x \geq 0\) and \(y \geq 0\)

Step 3: Finding common solution area

Look for the region in the plot where all the inequalities overlap. This overlapping region represents the solution set of the system of inequalities. It's the set of all (x, y) that satisfy all four inequalities at the same time.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting all possible solutions to an inequality on a plane. It is an essential skill in algebra which helps students understand how inequalities compare to equations. Rather than a single curve or line like you would have with an equation, inequalities fill an area on the graph.

• First, convert inequalities to equations to find boundary lines.
• Next, determine if the line should be solid (inclusive, for \(\leq\) or \( \geq \)) or dashed (exclusive, for \(<\) or \(>\)).
• Finally, shade the area of the graph where the inequality holds true.

This process provides a visual representation of all points that satisfy the inequality, making it easier to find where multiple inequalities intersect.

First Quadrant

Understanding the first quadrant is crucial when solving systems of inequalities, especially when dealing with problems related to graphing on the Cartesian plane. The first quadrant is defined as the section where both the x-coordinate and y-coordinate are positive.

• On a standard Cartesian plane, the first quadrant is the upper-right section.
• Points located here satisfy the conditions \(x \geq 0\) and \(y \geq 0\).
• This area includes the x and y-axes, but only where they are non-negative.

By focusing on this quadrant, we limit the graph to where both variables are positive, which can be particularly relevant in real-world scenarios where negative values are not applicable.

Inequality Solutions

Finding solutions to a system of inequalities requires identifying where shaded regions of each inequality overlap. Each system represents a set of criteria that solutions must meet. To find these, follow these steps:

• Graph each inequality on the same set of axes.
• Pay close attention to constraints marked by dashed or solid lines.
• Identify and shade the relevant regions of the graph for each inequality.
• Finally, determine the overlap of all shaded areas.

This common overlapping area is the solution to the system of inequalities, highlighting all possible values of \((x, y)\) that satisfy the system.

Cartesian Plane

The Cartesian plane is a two-dimensional graphing system essential in algebra, named after the French mathematician René Descartes. It's a tool for visualizing mathematical concepts and is fundamental for graphing equations and inequalities.

• It consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).
• These axes divide the plane into four quadrants, which help in organizing information.
• Each point on the plane is given by an ordered pair \((x, y)\).

Using the Cartesian plane, students can efficiently solve complex algebraic problems by translating abstract equations into tangible visuals. This visualization facilitates understanding where solutions exist within the plane's framework.