Problem 29 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 2 x-y>4 \\ 2 x-y>0 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region below both lines, \(y = 2x\) and \(y = 2x - 4\).

Step 1: Express Each Inequality

The given system of inequalities is \(2x - y > 4\) and \(2x - y > 0\). Start by expressing each inequality in the form \(y < mx + c\). For the first inequality, solve for \(y\):\[ y < 2x - 4 \]. Similarly, for the second inequality, solve for \(y\):\[ y < 2x \].

Step 2: Graph the Boundary Lines

Both inequalities \(y < 2x - 4\) and \(y < 2x\) suggest that we should graph the lines \(y = 2x - 4\) and \(y = 2x\). These lines should be dashed on the coordinate plane because the inequalities are strict (greater than, not greater than or equal to).

Step 3: Shade the Appropriate Regions

For the inequality \(y < 2x - 4\), shade the region below the line \(y = 2x - 4\). For \(y < 2x\), shade the region below the line \(y = 2x\). The solution set of the system is where both shadings overlap.

Step 4: Determine the Intersection

Identify the region on the graph where the two shaded areas overlap. This area represents the solution set for the system of inequalities. It lies below the line \(y = 2x\) but above \(y = 2x - 4\) between the points where these lines intersect any axes.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing these inequalities on a coordinate plane to visualize solutions. When we graph an inequality such as

• \( y < 2x - 4 \)
• \( y < 2x \)

we first change the inequality to an equation (by using \( = \)) to determine the boundary line. In the case of strict inequalities like \( < \), we use dashed lines to indicate that the points on the line are not included in the solution set. After plotting these lines, it’s crucial to determine which side of the line represents the solution. For example, for \( y < 2x - 4 \), shade the area below the line as this illustrates all the points where \( y \) values are less than \( 2x - 4 \). Similarly, do this for the other inequality as well. Graphing these lines helps identify where the solutions lie.

Solution Set

The solution set for a system of inequalities is the set of all points on the coordinate plane that satisfy all inequalities in the system. To find this set, we need to locate the region where the solutions to each individual inequality overlap. When tackling a problem like \[\begin{align*}2x - y & > 4 \2x - y & > 0\end{align*}\]you transform these into forms suitable for graphing and then determine the areas corresponding to each inequality.Once identified, these regions are shaded, and their intersection (or overlap) gives us the solution set. This ensures that every point within this shaded intersection satisfies both inequalities, giving us a comprehensive understanding of the possible solutions.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane corresponds to an ordered pair \((x, y)\), describing its location.Graphing systems of inequalities on this plane involves plotting lines which are derived from converting inequalities to equations (e.g., \(2x - y = 4\) and \(2x - y = 0\)).

• The x-axis represents horizontal positions.
• The y-axis represents vertical positions.

By interpreting the solutions graphically on this plane, we can easily see where the inequalities' conditions are met. This allows for a visual approach to understanding which regions are included in the system's solution set.

Intersection of Regions

The intersection of regions, in the context of systems of inequalities, is where the solution areas of two or more inequalities overlap. This intersection signifies that the conditions given by all inequalities are simultaneously satisfied.In our example, after shading the regions for

• \( y < 2x - 4 \)
• \( y < 2x \)

the overlapping area is crucial. It represents points where both conditions hold true. Finding this intersection involves closely analyzing and identifying the common space between the shaded regions.This shared area is the comprehensive solution set, showing all the (x, y) pairs that fulfill every inequality condition in the problem.