Problem 34 - Algebra A Combined Function

Question

Graph each system. $$ \left\\{\begin{array}{l} x-y<-1 \\ 4 x-3 y>0 \\ y>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the region of the coordinate plane where all three inequalities intersect, considering the respective half-planes.

Step 1: Graph the First Inequality

The first inequality given is \( x - y < -1 \). Start by graphing the equation \( x - y = -1 \) as a dashed line. To do this, find the intercepts: when \( x = 0 \), \( y = 1 \) and when \( y = 0 \), \( x = -1 \). Plot these points and draw a dashed line through them. Since the inequality is \(<\), shade below the line.

Step 2: Graph the Second Inequality

The second inequality is \( 4x - 3y > 0 \). Graph it by first converting it to the equation \( 4x - 3y = 0 \) (a dashed line because it is \(>\)). For intercepts: when \( x = 0 \), \( y = 0 \) and vice versa, which both give \((0,0)\). It's not sufficient alone, so find another point by setting \( x = 1 \), then \( y = \frac{4}{3} \). Plot these points, draw the line, and shade above this line since the inequality is \(>\).

Step 3: Graph the Third Inequality

The third inequality \( y > 0 \) represents the area above the x-axis. Shade above the x-axis as this inequality indicates any point with positive \( y \)-values.

Step 4: Identify the Solution Region

The solution region is the area that satisfies all inequalities simultaneously. It is the overlapping section of the shaded regions from the previous steps. As each inequality represents a half-plane, the solution will be the common region shared by them.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing the solutions to an inequality on a coordinate plane. Unlike equations, which give you a precise line or curve, inequalities represent a region of the plane that includes either one side of a line or an entire quadrant.

• Begin by turning the inequality into an equation to find the boundary line.
• Decide if the line should be solid or dashed. Use a dashed line for "<" or ">" and a solid line for "≤" or "≥".
• Determine whether to shade above or below the line based on the inequality sign.

Graphing these inequalities helps visualize which regions of the plane are included in the solution set.

Solution Region

The solution region for a system of inequalities is where the shaded areas of each individual inequality overlap. This common region represents all possible solutions that satisfy all of the given inequalities simultaneously.
Finding the solution region involves these steps:

• Graph each inequality as described previously, shading the appropriate side of each boundary line.
• Identify the area where the shaded parts intersect. This is your solution region.
• Ensure the solution region extends enough to cover all possibilities, especially when dealing with open boundaries (dashed lines).

This approach ensures we find all the potential solutions that meet the criteria set by each inequality.

Shading Regions

Shading regions on a graph helps to represent the solutions to an inequality visually. The shaded part indicates all the points that satisfy the inequality.
Here's how to determine where to shade:

• For "<" or ">", draw a dashed line and choose one side.
• For "≤" or "≥", draw a solid line and include the line in the shaded area.
• Use a test point, such as \((0,0)\), to verify which side of the line to shade. Substitute this point into the inequality. If it makes the inequality true, shade that side; if false, shade the opposite.

Shading correctly is crucial for accurately representing all possible solutions of an inequality.

Intercepts of Lines

Intercepts are critical points where the line crosses the axes on a graph. Finding the intercepts helps to graph the line accurately.
Here's how to find them:

• x-intercept: Set \(y = 0\) in the equation and solve for \(x\).
• y-intercept: Set \(x = 0\) and solve for \(y\).
• Plot these intercepts on the graph to start drawing your line.

Knowing the intercepts allows you to draw the initial line of the inequality before determining the shading, ensuring that the representation is both accurate and precise.