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Chapter 17: Problem 34

Graph the given inequalities on the number line. $$-4

Short Answer

Expert verified
Graph between -4 and -2 with open circles at each endpoint.

Step by step solution

01

Identify Endpoint Values

To graph the inequality \(-4 < y < -2\), begin by identifying the two endpoint values from the inequality, which are \(-4\) and \(-2\). These values indicate the boundary of where \(y\) can lie on the number line.
02

Place Open Circles

Since the inequality is strict (\(<\) rather than \(\leq\)), we do not include the endpoint values themselves. Use open circles to represent the endpoint values, placing one circle at \(-4\) and another at \(-2\) on the number line.
03

Shade the Interval

Shade the region on the number line between the open circles at \(-4\) and \(-2\). This shading indicates that \(y\) includes all the values that are greater than \(-4\) and less than \(-2\), but not \(-4\) or \(-2\) themselves.

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Key Concepts

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

Understanding the Number Line
A number line is a visual tool used to represent numbers in a straight line, where each point corresponds to a real number. It helps us easily compare values and see their relationships.

Here's how to get familiar with using a number line:
  • The line usually spans from negative to positive numbers, with zero typically at the center.
  • Negative numbers are to the left of zero and positive numbers to the right.
  • Each point on the number line represents a real number, and the distance between any two points measures the difference in their values.
For the inequality \(-4 < y < -2\), you locate \(-4\) and \(-2\) on the line. The numbers are to the left of zero since they are negative.

By getting comfortable with placing and interpreting values on this line, solving inequalities becomes much simpler.
Understanding Strict Inequalities
Strict inequalities, like \(<\) and \(>\), indicate that the values do not include the endpoints themselves. This is different from inclusive inequalities, such as \(\leq\) and \(\geq\), which do include those values.

When dealing with strict inequalities:
  • The inequality \(-4 < y < -2\) means \(y\) can be any number between \(-4\) and \(-2\), but precisely not \(-4\) nor \(-2\) themselves.
  • This is symbolized by open circles on the number line at those endpoints, signaling they are not part of the solution set.
Strict inequalities are crucial when determining ranges, as they explicitly define which values are included or excluded, allowing for accurate representations in both graph and solution.
Graphing Techniques for Inequalities
Graphing inequalities on a number line involves a few straightforward steps, allowing anyone to depict the range of solutions visually.

Let's break down the process:
  • Identify the boundary numbers involved, like \(-4\) and \(-2\) for \(-4 < y < -2\).
  • Place open circles on the number line at these points.
    This represents that these endpoints are not included in the solution.
  • Shade the region between the endpoints, showing the range of possible values for \(y\).
This visual method makes it clear what the solution set includes and excludes. It simplifies understanding complex mathematical concepts by translating them into easy-to-view images. Practicing these graphing techniques can significantly improve mathematical intuition, especially in terms of comprehending how inequalities function.

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Most popular questions from this chapter

Answer the given questions by solving the appropriate inequalities. A plane takes off from Winnipeg and flies due east at \(620 \mathrm{km} / \mathrm{h}\). At the same time, a second plane takes off from the surface of Lake Winnipeg \(310 \mathrm{km}\) due north of Winnipeg and flies due north at \(560 \mathrm{km} / \mathrm{h}\). For how many hours are the planes less than \(1000 \mathrm{km}\) apart?

Draw a sketch of the graph of the given inequality. $$y \leq 15-3 x$$

Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check. $$\frac{x}{x+3} > 1$$

Solve the given linear programming problems. A manufacturer produces a business calculator and a graphing calculator. Each calculator is assembled in two sets of operations, where each operation is in production 8 h during each day. The average time required for a business calculator in the first operation is 3 min, and 6 min is required in the second operation. The graphing calculator averages 6 min in the first operation and 4 min in the second operation. All calculators can be sold; the profit for a business calculator is 8 dollars, and the profit for a graphing calculator is 10 dollars. How many of each type of calculator should be made each day in order to maximize profit?

Use inequalities involving absolute values to solve the given problems. The production \(p\) (in barrels) of oil at a refinery is estimated at \(2,000,000 \pm 200,000 .\) Express \(p\) using an inequality with absolute values and describe the production in a verbal statement.
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