Problem 6 - Algebra A Combined Function

Question

Graph the solutions of each system of linear inequalities.. $$ \left\\{\begin{array}{l} y \leq 2 x+4 \\ y \geq-x-5 \end{array}\right. $$

Step-by-Step Solution

Answer

Shade the overlapping region of the two inequalities, below \(y = 2x+4\) and above \(y = -x-5\).

Step 1: Understand the Inequalities

There are two inequalities to consider: \( y \leq 2x + 4 \) and \( y \geq -x - 5 \). These inequalities describe shaded regions on the coordinate plane, bounded by their respective lines.

Step 2: Plot the Boundary of \(y \leq 2x + 4\)

The line \( y = 2x + 4 \) is a straight line with a slope of 2 and a y-intercept of 4. To plot, start at (0, 4) on the y-axis and use the slope to find another point, such as going right 1 unit and up 2 units, reaching (1, 6). Draw the line. Since it's \( y \leq 2x + 4 \), shade below the line.

Step 3: Plot the Boundary of \(y \geq -x - 5\)

The line \( y = -x - 5 \) is a straight line with a slope of -1 and a y-intercept of -5. Start at (0, -5) on the y-axis and use the slope to find another point, such as going right 1 unit and down 1 unit, reaching (1, -6). Draw the line. Since it's \( y \geq -x - 5 \), shade above the line.

Step 4: Identify the Solution Region

The solution to the system of inequalities is the overlap of the two shaded regions on the graph. This is the area where both conditions \( y \leq 2x + 4 \) and \( y \geq -x - 5 \) are satisfied simultaneously. It is a region that forms an overlapping area typically forming a quadrilateral.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface that allows us to graph equations and inequalities. It consists of two perpendicular number lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis.
The point where the axes intersect is known as the origin, labeled as (0, 0). To plot a point on this plane, you follow an order: first, move along the x-axis, and then the y-axis.
This grid system is fundamental in representing relationships between two variables, like these inequalities.

System of Inequalities

A system of inequalities includes two or more inequalities that you consider simultaneously. In this exercise, the system consists of:

• \( y \leq 2x + 4 \)
• \( y \geq -x - 5 \)

Each inequality describes a region of solutions. A solution for the system is a point that satisfies all inequalities at the same time. Understanding how to work with these helps in many fields, including economics and engineering.
Visualizing these inequalities on the coordinate plane can make finding possible solutions more straightforward.

Shaded Regions

When graphing linear inequalities, the inequality's border is shown as a line. The inequality sign tells us whether to shade above or below this line. This indicates all points that satisfy the inequality.
For instance, if you have the inequality \( y \leq 2x + 4 \), your line will indicate the border, and you will shade below it.
The shaded region represents all the points where the inequality holds true.

• "\( y \leq 2x + 4 \) is shaded below the line.
• "\( y \geq -x - 5 \) is shaded above the line.

This provides a visual representation of potential solutions to individual inequalities.

Solution Region

The solution region is the area on the coordinate plane where the shaded regions of all inequalities overlap. This represents all points that satisfy each inequality in the system simultaneously.
In the context of our inequalities, it is where both following conditions are true:

• "\( y \leq 2x + 4 \)
• "\( y \geq -x - 5 \)

This overlap forms a particular shape, often a polygon, like a quadrilateral. Determining this area is crucial, as it is the comprehensive set of solutions to the system. Make sure to interpret the intersection accurately to determine the solution region.