Problem 34 - A Graphical Approach to Precalculus with Limits

Question

When graphing \(y>3 x-6,\) would you shade above or below the line \(y=3 x-6 ?\) Explain your answer.

Step-by-Step Solution

Answer

Shade above the line \( y = 3x - 6 \).

Step 1: Understand the Inequality

We are given the inequality \( y > 3x - 6 \). This inequality means that for any point that satisfies it, the y-coordinate of the point must be greater than the value given by the expression \( 3x - 6 \).

Step 2: Graph the Line

To solve inequalities graphically, start by graphing the related equation \( y = 3x - 6 \). This will be a straight line with a slope of 3 and a y-intercept of -6.

Step 3: Choose a Test Point

Select a test point not on the line, such as the origin (0,0), to determine which side of the line to shade. Substitute the test point into the inequality \( y > 3x - 6 \).

Step 4: Substitute the Test Point

Substituting (0,0) in the inequality: \( 0 > 3(0) - 6 \) simplifies to \( 0 > -6 \), which is true. Therefore, the region that contains the point (0,0) should be shaded.

Step 5: Shade the Correct Region

Because the test point (0,0) is below the line and satisfies the inequality, shade the area above the line \( y = 3x - 6 \) on the graph.

Key Concepts

Linear Equations

A linear equation is a foundational concept in mathematics that forms a straight line when graphed on the coordinate plane. It has the general form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is where the line crosses the y-axis.

For example, in the exercise with the inequality \( y > 3x - 6 \), the corresponding linear equation \( y = 3x - 6 \) has a slope of 3 and a y-intercept of \(-6\). This means for each unit increase in \( x \), \( y \) increases by 3 units. The line starts at point (0, -6) on the y-axis and rises sharply across the plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph equations of all kinds, including linear equations and inequalities. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Each point on this plane is defined by a pair of numbers \((x, y)\), which are its coordinates.

Graphing on the coordinate plane allows us to visually analyze relationships and solutions. For a linear equation like \( y = 3x - 6 \), plotting points that satisfy the equation helps us draw a precise line. The plane enables a clear visualization of where one region lies relative to a graphed line - crucial for understanding inequality solutions.

Inequality Solution

An inequality like \( y > 3x - 6 \) represents a set of points that fulfill a certain condition, differentiating it from an equation which aligns points exactly on a line. The inequality indicates that we are searching for all points where the \( y \)-coordinate is greater than \( 3x - 6 \).

To solve graphically, first plot the line \( y = 3x - 6 \) but use a dashed line to show that points on the line are not part of the solution (since the inequality is strict). Next, determine which region to shade based on a test point. The shaded region - in this problem, above the line - contains all the points that satisfy the inequality.

Test Points

Using a test point is a simple, effective way to determine which side of a line to shade when graphing inequalities. Choose a point that does not lie on the line, ideally one that makes calculations easy. For our inequality \( y > 3x - 6 \), the test point (0,0) is often a convenient choice because substituting it into the inequality provides clear insight.

When you substitute \((0,0)\) into the inequality and find that \(0 > -6\) holds true, it shows that the area containing (0,0) satisfies the inequality. As a result, this helps decide which region on the graph to shade, confirming that all points in that area meet the condition specified by the inequality.