Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region where the solution to both inequalities overlap, which forms a polygonal region bounded by the lines \(y = -x + 4\) and \(y = 2x - 4\).

Step 1: Analyze and plot the first inequality

Let's first analyze the inequality \(x+y \leq 4\). This inequality can be rewritten as \(y \leq -x + 4\) which represents a line with a negative slope and y-intercept at 4. The inequality solution is all points below or on this line. This line can be graphed by plotting the y-intercept at 4 and using the slope of -1 to find other points on this line.

Step 2: Analyze and plot the second inequality

Now let's analyze the second inequality \(y \geq 2x -4\). This inequality represents a line with slope 2 and y-intercept at -4. The inequality solution is every point above or on this line. This line can be graphed by plotting the y-intercept at -4 and applying the slope of 2 to find other points on this line.

Step 3: Find the solution for this system of inequalities

The solution to the system of inequalities is where the solutions to both inequalities overlap. Since these are linear inequalities, these solutions will form a region on the graph. By observing the graph, one can identify the overlapped area which is the solution to the system of inequalities.

Key Concepts

Graphing Inequalities

When graphing inequalities, the first step is to treat them similarly to equations, with the exception that we are looking for a range of solutions rather than a single line. Each inequality will initially be plotted as if it were an equation, using a solid or dashed line. A solid line indicates that points on the line are included in the solution (representing "less than or equal to" or "greater than or equal to"), while a dashed line means the points on the line are not included (just "less than" or "greater than").

For example, for the inequality \(x + y \leq 4\), we graph the line \(y = -x + 4\) which involves:

• Locating points like the intercept where \(y = 4\).
• Using the slope \(-1\) to plot additional points.

The shading will be below this line because of the "less than or equal to" sign, indicating that any point below or on the boundary line satisfies the inequality.Now consider \(y \geq 2x - 4\). Here, the line \(y = 2x - 4\) is plotted with:

• A y-intercept at \(-4\).
• A slope of \(2\), guiding the direction of additional points.

The shading will be above this line, reflecting "greater than or equal to". Points above or on this line are part of the solution.

Linear Inequalities

Linear inequalities look like linear equations but with inequality symbols (\(\leq, \geq, <, >\)) instead of the equal sign. Understanding their basic characteristics helps us in plotting and interpreting graphs.

Each linear inequality defines a half-plane, a part of the graph that contains either every point above or below a boundary line. The boundary line separates these regions and represents all the points where the linear equation \(f(x, y) = 0\) holds true.

For the inequality \(x + y \leq 4\), substituting it for equality gives \(x + y = 4\), describing a line with a slope and y-intercept. This line is a boundary for the inequality. Everything below it (including the line itself) satisfies the inequality. Likewise, for \(y \geq 2x - 4\), the boundary is the line \(y = 2x - 4\), and everything above or on this line makes the inequality true.

Linear inequalities often make use of:

• Slope-field techniques, allowing us to interpret the direction and location for shading within the graph.
• Analysis of the intercepts to identify specific points of the line.

This graphical understanding articulates where solutions are legitimate on the coordinate plane.

Solution Set of Inequalities

In solving systems of inequalities, we aim to find the common solution area that satisfies all the inequalities involved. This area is the overlap of all the shaded regions from individual inequalities, forming a feasible region.

For our given system:

• \(x + y \leq 4\) covers a region below or on the line \(y = -x + 4\).
• \(y \geq 2x - 4\) highlights a zone above or on the line \(y = 2x - 4\).

The solution set is identified where these two regions intersect. This means finding the combined area that lies both below the first line yet above the second line, creating a defined polygonal region. All points within this intersection satisfy both inequalities.

The boundary lines themselves may or may not be included based on the equality within the inequalities. This overlapped intersection is the most critical part in graphing solutions, indicating viable answers to the system laid out.