Problem 27 - Precalculus Mathematics for Calculus

Question

Sketch the region given by the set. $$\\{(x, y) | 1

Step-by-Step Solution

Answer

The region is a vertical strip between \( x = 1 \) and \( x = 2 \), excluding the lines.

Step 1: Understand the Set Description

The set \( \{(x, y) | 1

Step 2: Draw Boundary Lines

Draw two vertical dashed lines at \( x = 1 \) and \( x = 2 \) on the coordinate plane. These lines are dashed to indicate the inequality bounds \( 1 < x < 2 \), showing that these x-values are not included in the solution.

Step 3: Highlight the Region

Shade the entire region between the dashed lines. This represents all points with \( x \) values between 1 and 2, regardless of the \( y \) value, covering the entire vertical strip.

Key Concepts

Inequality Representation

When it comes to understanding inequalities, particularly on a coordinate plane, representation becomes key. Inequality representation refers to expressing conditions that specify a particular set of values. In the given exercise, the inequality \(1 < x < 2\) tells us about the range of x-coordinates allowed.

The inequality \(1 < x < 2\) means that x-values must be greater than 1 and less than 2. The signs "<" and ">" (less than and greater than) imply that the boundaries themselves are not included. In other words, x cannot be exactly 1 or 2, but it can be any value very slightly above 1 until just before 2.

Understanding this helps you map out which parts of the coordinate plane are relevant. It is essential for visualizing and drawing the corresponding area that satisfies these conditions. Whenever you graph such inequalities, you're identifying where on the plane these conditions are true.

Graphical Shading

Graphical shading is the technique used to visually represent solutions to inequalities on a coordinate plane. After determining the inequality, the next step in graphing is to shade the region that corresponds to the solution.

In the context of the exercise, once you identify the x-values between 1 and 2, shading confers life to this abstract representation. With inequalities, you don't always have a single line or curve. Instead, you have a region that needs to be highlighted. Here, you would shade the entire region between the dashed lines representing \(x = 1\) and \(x = 2\).

• This shaded region extends vertically without bound, since there are no limits on y-values.
• Shading also allows others viewing the graph to instantly understand which area is included in the solution.

Choosing to shade the region appropriately clarifies the solution set at a glance, making it easier to grasp and communicate. It also prevents errors in interpretation and reinforces the constraints of the inequality.

Boundary Lines

Boundary lines are important features when graphing inequalities, as they mark the edges of the region that satisfy the inequality conditions. In this exercise, boundary lines are drawn at \(x = 1\) and \(x = 2\).

These lines are dashed instead of solid because the condition specified in the inequality is strictly less than or greater than. This indicates that while these lines help define the boundaries, points exactly on these lines are not included in the solution set. You visualize them as the cutoff markers.

• Always use dashed lines when the inequality symbol is "<" or ">".
• If the inequality included an "equal to" component, you would use a solid line to include the boundary.

Boundary lines are a crucial part of graphically representing inequalities since they provide a clear visual delineation of what the solution set does and does not include. This makes interpreting and shading the correct region much simpler.

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Other exercises in this chapter

Problem 27

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$2 \leq x+5

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Problem 27

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Problem 27

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Problem 27

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$(t-4)^{2}=(t+4)^{2}+32$$

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Key Concepts

Inequality Representation

When it comes to understanding inequalities, particularly on a coordinate plane, representation becomes key. Inequality representation refers to expressing conditions that specify a particular set of values. In the given exercise, the inequality \(1 < x < 2\) tells us about the range of x-coordinates allowed.

The inequality \(1 < x < 2\) means that x-values must be greater than 1 and less than 2. The signs "<" and ">" (less than and greater than) imply that the boundaries themselves are not included. In other words, x cannot be exactly 1 or 2, but it can be any value very slightly above 1 until just before 2.

Understanding this helps you map out which parts of the coordinate plane are relevant. It is essential for visualizing and drawing the corresponding area that satisfies these conditions. Whenever you graph such inequalities, you're identifying where on the plane these conditions are true.

Graphical Shading

Graphical shading is the technique used to visually represent solutions to inequalities on a coordinate plane. After determining the inequality, the next step in graphing is to shade the region that corresponds to the solution.

In the context of the exercise, once you identify the x-values between 1 and 2, shading confers life to this abstract representation. With inequalities, you don't always have a single line or curve. Instead, you have a region that needs to be highlighted. Here, you would shade the entire region between the dashed lines representing \(x = 1\) and \(x = 2\).

• This shaded region extends vertically without bound, since there are no limits on y-values.
• Shading also allows others viewing the graph to instantly understand which area is included in the solution.

Choosing to shade the region appropriately clarifies the solution set at a glance, making it easier to grasp and communicate. It also prevents errors in interpretation and reinforces the constraints of the inequality.

Boundary Lines

Boundary lines are important features when graphing inequalities, as they mark the edges of the region that satisfy the inequality conditions. In this exercise, boundary lines are drawn at \(x = 1\) and \(x = 2\).

These lines are dashed instead of solid because the condition specified in the inequality is strictly less than or greater than. This indicates that while these lines help define the boundaries, points exactly on these lines are not included in the solution set. You visualize them as the cutoff markers.

• Always use dashed lines when the inequality symbol is "<" or ">".
• If the inequality included an "equal to" component, you would use a solid line to include the boundary.

Boundary lines are a crucial part of graphically representing inequalities since they provide a clear visual delineation of what the solution set does and does not include. This makes interpreting and shading the correct region much simpler.