Problem 4 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$2 x-y>4$$

Step-by-Step Solution

Answer

The solution to the given inequality \(2x - y > 4\) is a graph of line where all the points above the line are shaded.

Step 1: Convert The Inequality Into An Equation

First step is to convert the inequality into an equation. It can be done by keeping the left side as it is and making the right side equal to zero. We get the equation \(2x - y = 4\)

Step 2: Graph The Equation

Next, graph the equation \(2x - y = 4\) on a 2 Dimensional plane. You can use any method such as calculating the intercepts or using slope and y-intercept. For this equation, The y-intercept is -4 and the slope is 2. Start by plotting the y-intercept at (0,-4). The slope is 2 which means for each step to the right on the x-axis, go 2 steps up on the y axis. Using this, you get a straight line.

Step 3: Shade The Appropriate Region

Now we look at the sign of the inequality. It is 'greater than', so we shade the area above the line of the equation.

Key Concepts

Two-Dimensional Plane

When you start graphing an inequality like \(2x - y > 4\), it's important to understand the concept of the two-dimensional plane. This plane is basically just a flat surface where you can plot points using two numbers: \(x\) and \(y\) coordinates.

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.

These axes intersect at the origin, which is the point \((0,0)\). Any point on this plane is located using these two coordinates, which tells you exactly where to place it.
Think of the two-dimensional plane like a giant sheet of graph paper laid out flat. Understanding this will help you visualize where to draw lines and where to shade regions when graphing inequalities.

Y-Intercept

The y-intercept is a crucial concept when graphing a linear equation or inequality. It is the point where the line crosses the y-axis.
In the equation \(2x - y = 4\), you can rearrange it to solve for \(y\) and find that the y-intercept is -4. This means the point \((0, -4)\) is where the line hits the y-axis.

• To find the y-intercept, set \(x = 0\) and solve for \(y\).
• The y-intercept provides a starting point for plotting your line.

Once you have this intercept, it's simple to start your graph.
Just plot the y-intercept on the vertical axis, and you're well on your way to completing the graph.

Slope

Slope is the measure of how steep a line is, and it plays an essential role in graphing.
The slope of a line can be found from the equation of the line in slope-intercept form \(y = mx + b\), where \(m\) is the slope. In our equation \(2x - y = 4\), the slope is 2.
This tells you how far to go up or down, as well as left or right, between two points on the line.

• In this example, a slope of 2 means that for every step you move to the right on the x-axis, you move 2 steps up on the y-axis.
• The sign of the slope tells you the direction: a positive slope goes up as you move right, while a negative slope would go down.

Understanding slope helps you accurately draw the line through the y-intercept and ensures that the line is correctly aligned on the two-dimensional plane.

Shading Regions

Shading is the step that brings inequality to life on a graph. Once you have drawn the line, you need to decide which side of the line to shade.
This helps indicate all the points where the inequality is true.

• For \(2x - y > 4\), the inequality is "greater than," which means you shade above the line.
• The direction of shading depends on the inequality sign: "greater than" usually means shading above or to the right, while "less than" means shading below or to the left.

Always remember:- Use a test point if you're unsure—choose any point not on the line, like the origin \((0,0)\). Plug it into the inequality. If it's true, shade that side. If false, shade the opposite side.
Shading the correct region is key to visualizing all solutions to the inequality on the two-dimensional plane.