Problem 14 Graph using the test point metho... [FREE SOLUTION]

Chapter 9: Problem 14

Graph using the test point method. \(y \geq 3 x\)

Short Answer

Expert verified

Graph the solid line \(y = 3x\) using points such as \((0,0)\) and \((1,3)\). Choose a test point like \((0,1)\) and plug it into the inequality \(y \geq 3x\). Since the test point satisfies the inequality, shade the region above the line \(y = 3x\) as the solution set.

Step by step solution

Graph the Line \(y = 3x\)

First, we need to graph the line \(y = 3x\). Since the inequality is "greater than or equal to," the line will be solid (not dashed) as it's included in the solution set. To graph the line, choose two points to plot. For example: - When \(x = 0\), \(y = 3(0) = 0\). So the point \((0,0)\) is on the line. - When \(x = 1\), \(y = 3(1) = 3\). So the point \((1,3)\) is on the line. Plot these points and draw a solid line through them.

Choose a Test Point

Now, we need to choose a test point to determine which side of the line represents the solution set. A good test point is the origin, \((0, 0)\), if it is not on the line. In this case, the origin is on the line, so we must choose a different point. A common choice is \((0, 1)\).

Test the Inequality with the Test Point

Plug in the chosen test point \((0, 1)\) into the inequality \(y \geq 3x\): \(1 \geq 3(0)\) \(1 \geq 0\) The inequality is true for this test point, so we will need to shade the region that includes the point \((0, 1)\).

Shade the Appropriate Region

Since the region that includes the test point is the solution set, shade the region above the line (where the test point is located). Make sure the line \(y = 3x\) is solid, as it is included in the solution set. The graph of \(y \geq 3x\) is now complete.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Test Point Method

The test point method is a technique used to graphically solve inequalities. This method helps determine which side of the boundary line contains the solutions to the inequality.
When applying this method:

First, graph the boundary line. In our case, it is the line associated with the equation, which is seemingly obtained by replacing the inequality with an equals sign, such as transforming \( y \geq 3x \) to \( y = 3x \).
Choose a test point not on the line (often the origin, \((0,0)\), if possible).
Substitute the test point into the inequality. If the inequality holds true with this test point, then the region containing this point is part of the solution set.

It鈥檚 a straightforward method to determine where the solutions lie, especially for linear inequalities.

Solution Set

The solution set represents all the possible values that satisfy the inequality. It is the entire region on the graph where the inequality holds true.
Here:

First, graph the boundary line (solid or dashed depending on whether the inequality includes equals).
Use the test point to determine which side of the boundary line contains the solution set.
Shade this region thoroughly to signify it includes all the solutions.

Another important aspect is how the boundary line works: if it is solid, it means solutions on the line are included, and if dashed, they are not. This dictates the nature of the solution set.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a line through them. In the example, the equation is \( y = 3x \).
To graph:

Choose values for \( x \) to find corresponding \( y \) values.
Plot these (\( x, y \)) points on a grid.
Draw a line through the points. Here, it鈥檚 solid because the inequality is ">=", meaning it includes the boundary line.

Accurate graphing is crucial for correctly identifying the solution set for inequalities. Practice plotting diverse points to ensure precision.

Inequality Symbols

Inequality symbols help us understand the nature of the relationship between two expressions. The main symbols include:

\( \geq \) (greater than or equal to): Indicates that the value on the left is either greater than or the same as the value on the right.
\( \leq \) (less than or equal to): Means the value is either less than or equal to.
\( > \) and \( < \): Indicate strictly greater or strictly less than.

For graphing inequalities, these symbols dictate whether the boundary line is included in the solution set (solid line for \( \geq \) or \( \leq \)) or not (dashed line for \( > \) or \( < \)). This distinction is vital to visually represent the set solutions accurately on the graph.

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91影视

Graph using the test point method. \(y \geq 3 x\)

Short Answer

Step by step solution

Graph the Line \(y = 3x\)

Choose a Test Point

Test the Inequality with the Test Point

Shade the Appropriate Region

Key Concepts

Test Point Method

Solution Set

Graphing Linear Equations

Inequality Symbols

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