Problem 12 - Precalculus with Limits

Question

In Exercises 5-18, sketch the graph of the inequality. $$y>4 x-3$$

Step-by-Step Solution

Answer

The region above the line \(y = 4x - 3\) (not including the line itself) is the solution to the inequality \(y > 4x - 3\).

Step 1: Sketching the Line

Begin by sketching the line associated with the inequality. This is the line \(y = 4x - 3\), which has a slope of 4 and intersects the y-axis at -3. As the inequality is \(y > 4x - 3\) and not \(y \geq 4x - 3\), it’s best to draw this line as a dashed or dotted line to indicate that points on the line are not included in the solution region.

Step 2: Choosing Test Point

Choose a test point that is not on the line. A good choice is often the origin \((0,0)\), as long as the line doesn't pass through it. In this case, the line does not pass through the origin, so it can be used as a test point.

Step 3: Testing the Inequality

Use the test point in the inequality and determine if it results in a true statement. If we put \(0\) in place of \(y\) and \(x\) in the inequality \(y > 4x - 3\), we get \(0 > -3\), which is true.

Step 4: Shading the Region

Since the test point results in a true statement, the region that includes \((0,0)\) should be shaded to indicate the solution area. This will be the area above the line when the inequality is \(y > 4x - 3\).

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, known as coordinates. These numbers are written as \(x, y\), where \x\ represents the horizontal position and \y\ represents the vertical position. The plane is divided into four quadrants by a horizontal axis (x-axis) and a vertical axis (y-axis). Understanding the coordinate plane is essential for graphing any mathematical equation or inequality.

• The x-axis runs left to right (horizontal).
• The y-axis runs up and down (vertical).
• The origin is the point \(0, 0\) where the axes intersect.

Points are plotted based on these coordinates, helping us visualize relationships expressed by equations and inequalities.

Using Test Points

Test points help determine which side of the boundary line represents the solution to an inequality. To use a test point, select a point on the coordinate plane that is not on the boundary line; often, the origin \(0, 0\) is a wise choice if it's not part of the line.

• Substitute the x and y values of the test point into the inequality.
• If the result is true, then the chosen side of the plane is the solution region.
• If the result is false, the solution lies on the opposite side.

For example, by testing the origin in the inequality \(y > 4x - 3\), we find it true since \(0 > -3\). This verifies that the region containing the origin is indeed part of the solution.

Graphing Linear Equations

A linear equation like \(y = 4x - 3\) represents a straight line on the coordinate plane. This particular line has a slope of 4, meaning it rises 4 units for every 1 unit it runs to the right. The y-intercept is -3, indicating where the line crosses the y-axis.

• Slope (m) tells us how steep the line is; here it's 4.
• Y-intercept (b) shows where it crosses the y-axis; here it's -3.
• For inequalities, use a dashed line for \(>\) or \(<\) indicating boundary points aren’t included.

These characteristics allow us to draw the line accurately and set the stage for analyzing the inequality.

Identifying the Solution Region

The solution region in an inequality graph is the area where the inequality holds true. After drawing the boundary line, determined by the corresponding linear equation, use shading to indicate which part of the plane includes the solutions.

• A dashed line means points on the line aren’t included.
• The side where the test point makes the inequality true is shaded.
• For \(y > 4x - 3\), the region above the dashed line is shaded.

Visualizing this solution region helps understand how inequalities differ from equations, as they describe ranges of solutions, not just specific points.