Problem 12 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{l}y+2<2 x \\\y-x>4\end{array}\right.$$

Step-by-Step Solution

Answer

Graph the inequalities \( y < 2x - 2 \) and \( y > x + 4 \). Shade below the first line and above the second line; the overlap is the solution.

Step 1: Solve the First Inequality for y

Start with the inequality \( y + 2 < 2x \). Subtract 2 from both sides to isolate \( y \), resulting in \( y < 2x - 2 \).

Step 2: Graph the Boundary for First Inequality

Graph the line \( y = 2x - 2 \) using a dashed line since the inequality is \( < \). This means that the points on the line itself are not included in the solution.

Step 3: Shade the Solution Region for First Inequality

Since the inequality is \( y < 2x - 2 \), shade the region below the line \( y = 2x - 2 \) because this represents the area where \( y \) is less than \( 2x - 2 \).

Step 4: Solve the Second Inequality for y

Now, take the inequality \( y - x > 4 \) and solve for \( y \) by adding \( x \) to both sides: \( y > x + 4 \).

Step 5: Graph the Boundary for Second Inequality

Graph the line \( y = x + 4 \) using a dashed line since the inequality is \( > \). Again, points on the line are not part of the solution.

Step 6: Shade the Solution Region for Second Inequality

Since \( y > x + 4 \), shade the region above the line \( y = x + 4 \), indicating where \( y \) is greater than \( x + 4 \).

Step 7: Identify the Overlapping Region

The solution to the system of inequalities is where the shaded regions from both inequalities overlap. This overlapping area represents all the solutions that satisfy both inequalities.

Key Concepts

System of Inequalities

When we talk about a system of inequalities, we're referring to two or more inequalities that are considered together. These inequalities are like equations that feature 'less than', 'greater than', or sometimes 'equal to' signs. In our example, we have:

• \( y + 2 < 2x \)
• \( y - x > 4 \)

The aim is to find all possible values of variables that satisfy both inequalities at the same time.
We can think of it as finding a solution set that meets the criteria of both inequalities. To solve, we graph both on a coordinate plane and see where their solution regions overlap.
This overlapping area showcases all the pairs of \((x, y)\) that satisfy both inequalities simultaneously.

Solution Region

The solution region is a key concept when graphing systems of inequalities. It refers to the part of the graph where all the shaded regions from each inequality intersect. This shaded overlap represents where values satisfy all inequalities in the system.
For instance, consider the inequalities from our exercise:

• For \( y < 2x - 2 \), the area shaded is below the boundary line of this inequality.
• For \( y > x + 4 \), the shaded area is above its boundary line.

The solution region is where these two shaded areas meet, as it's the set of all points \((x, y)\) that work for both conditions. This region is generally a polygon or an unbounded area, depending on the inequalities involved.
Visually identifying this intersection is crucial for solving systems of inequalities.

Boundary Lines

Boundary lines play an essential role when graphing inequalities. They're the lines you graph that correspond to each inequality after converting it for graphing purposes. Let's break this down:

• Each inequality has an associated line; for this system, they are:
• \( y = 2x - 2 \)
• \( y = x + 4 \)
• These lines help define where the inequality's solution region starts and ends.

For inequalities using < or >, such as ours, we use dashed lines. The dashed line indicates that points on the boundary itself are not included in the solution set. However, if you encounter \( \leq \) or \( \geq \), you would use a solid line to also include those points.
These boundary lines help define the limits of where the inequality's solution exists.

Inequality Graphing Steps

Graphing inequalities involves several clear steps. Here's a simplified process you can refer back to:

Step 1: Solve for y

• Both inequalities should be solved for \( y \). In our exercise, turning \( y + 2 < 2x \) into \( y < 2x - 2 \), and \( y - x > 4 \) to \( y > x + 4 \).

Step 2: Graph the Boundary Line

• Draw the lines representing each inequality. Use dashed lines for < or > inequalities.

Step 3: Shade the Solution Region

• Shade the area above or below each line depending on the direction of the inequality sign. "Less than" means below; "greater than" means above.

Step 4: Identify Overlap

• The last step is to find where your shaded regions overlap. This overlap is where the solution to the entire system is found.

Following these steps ensures a methodical approach to graphing systems of inequalities and accurately identifying the solution region.