Problem 24 - Elementary Algebra

Question

For the following problems, graph the equation of inequality. $$ 2 x+3 y>6 $$

Step-by-Step Solution

Answer

Question: Graph the inequality \(2x + 3y > 6\). Answer: The graph of the inequality \(2x + 3y > 6\) has a dashed boundary line given by \(y = \frac{6-2x}{3}\). Since the origin (0,0) does not satisfy the inequality, we shade the side of the line opposite to the test point (0,0) to represent the region that satisfies the inequality.

Step 1: Find the boundary line

To find the boundary line, we need to turn the inequality into an equation. We can do this by replacing the inequality symbol (>) with an equal sign (=). This gives us the equation of the boundary line: $$ 2x + 3y = 6 $$

Step 2: Solve the boundary line for y

Let's solve the equation for \(y\) to make it easier to graph the line: $$ y = \frac{6-2x}{3} $$ Now we can identify the line \(y = \frac{6-2x}{3}\) as the boundary line.

Step 3: Determine which side to shade

Now we need to find out which side of the line to shade to represent the inequality. To do this, we can pick a test point that is not on the line and plug it into the inequality. If the test point satisfies the inequality, we will shade the side of the line that includes the test point. Otherwise, we will shade the opposite side. Let's use the origin (0,0) as our test point, because it's an easy point to work with. Plugging the coordinates into the inequality, we get: $$ 2(0) + 3(0) > 6 $$ $$ 0 > 6 $$ Since 0 is not greater than 6, the test point does not satisfy the inequality. This means that we need to shade the side of the line opposite to the test point.

Step 4: Graph the shaded region

To graph the inequality, draw the line \(y = \frac{6-2x}{3}\) on the coordinate plane. Since the inequality is strictly greater than (>) and not greater than or equal to (≥), the boundary line should be drawn as a dashed line to indicate that the points on it are not included in the solution set. Next, shade the side of the line opposite to the test point (0,0) to represent the region that satisfies the inequality \(2x + 3y > 6\). And that is your graph of the inequality.

Key Concepts

Boundary Line

When graphing linear inequalities, the first step is to establish the boundary line. This line separates the coordinate plane into two distinct regions. To find this boundary line, we convert the inequality into an equation. For our exercise, the inequality \(2x + 3y > 6\) is changed into the equation \(2x + 3y = 6\). This equation forms a straight line.

• The boundary line itself acts as a border for the shaded area that represents all possible solutions to the inequality.
• In our case, because the inequality symbol is 'greater than' and not 'greater than or equal to', the boundary line is drawn with dashed marks.

This dashed marking indicates that points on the line do not satisfy the inequality but rather form its boundary.

Test Point

After determining the boundary line, the test point helps decide which side of the line to shade. A test point is a reference used to check which side of the boundary line holds the solutions to the inequality. Usually, the origin point (0,0) is preferred, unless it is on the line itself, due to its simplicity.

• For our inequality \(2x + 3y > 6\), we test the origin by substituting \(x = 0\) and \(y = 0\) into the inequality, which gives us \(0 > 6\).
• Since this is false, the origin does not satisfy the inequality. Thus, the side of the line that includes the origin will not be shaded.

The choice of a test point helps verify the correct shaded region, ensuring that only the side which fulfills the inequality is marked.

Shaded Region

The shaded region in the graph of an inequality represents all the solutions that satisfy the given inequality. After determining the boundary line and using a test point, shading is the step where you visibly depict the solution region.

• For the inequality \(2x + 3y > 6\), once we know the origin does not satisfy the inequality, the region not containing the origin is shaded.
• This shaded area includes all the coordinate points that make the inequality true, marking them as solutions.

Remember, the shaded region is exclusive of the boundary line due to the dashed line indicating points on the boundary are not part of the solution set.

Coordinate Plane

The coordinate plane is an essential element for graphing inequalities and consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. When plotting an inequality, the coordinate plane serves as the canvas displaying all possible solutions.

• Each point on the coordinate plane is defined by an (x, y) pair, which indicates its position in terms of horizontal and vertical displacement from the origin (0,0).
• A proper understanding of the coordinate plane allows for accurate graphing, helping to appropriately place the boundary line and identify which regions satisfy the inequality.

The use of a coordinate plane is crucial as it organizes the values, simplifies the process of drawing lines, and highlights the relationship between the points and the inequality condition.