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Chapter 14: Problem 53

Graph the set. (GRAPH CANNOT COPY) $$\\{x |-3

Short Answer

Expert verified
The graph of the inequality \(-3 < x < 3\) is a line segment on the number line that goes from -3 to +3, excluding the end points at -3 and +3. The open circles at -3 and +3 mark the boundaries of the solution set, and the shaded region between these points represents all the values of x that satisfy the inequality.

Step by step solution

01

Analyze the inequality

The inequality given is \(-3 < x < 3\). This inequality can be broken down into two parts: x is greater than -3 and x is less than 3. Note that x cannot be equal to -3 or 3 itself, but any value in between.
02

Marking points on the number line

Plot a number line, and mark two dots at -3 and +3. These dots represent the two boundaries of our solution set. Since x cannot be equal to -3 or 3, make sure to use open circles to indicate these points.
03

Shade the solution set

Since x must be greater than -3 and less than 3, shade the region between these two points to indicate the solution set.
04

Check for correctness

To check whether the graph is correct, pick a point inside the shaded region, such as 0. Substituting this into the original inequality, you should find that the inequality holds true. This verifies that the solution set has been represented correctly on the graph.

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Key Concepts

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

Number Line
A number line is a visual tool that helps us understand numerical relationships. It's a horizontal line that represents real numbers as points along its length. To graph \(-3 < x < 3\), first draw a simple horizontal line.
Then, locate the points \(-3\) and \(3\) on this line. These points will serve as boundaries for our solution. A number line is especially useful when dealing with inequalities as it provides a clear, visual representation of which numbers fall into the range indicated by the inequality.
Keep your number line neat and spaced evenly to maintain clarity, making it easier to interpret the solution graphically.
Inequalities
Inequalities express the relationship between two values, showing that one is larger or smaller than the other. In this exercise, we have a compound inequality: \(-3 < x < 3\). This means that the variable \(x\) is greater than \(-3\) and less than \(3\).
It's important to remember that inequalities can include other symbols like \(>\), \(\leq\), and \(\geq\), which specify different relationships. Here, the absence of equality means \(x\) cannot be \(-3\) or \(3\), but any value in between. Understanding the type of inequality is crucial to graphing it correctly on a number line.
Solution Set
The solution set is the range of values that satisfy the given inequality. For \(-3 < x < 3\), the solution set includes every possible number between \(-3\) and \(3\), but not the endpoints themselves.
In a visual context, this range is represented by shading the portion of the number line that lies between \(-3\) and \(3\). When graphing, the shaded area clearly shows where all potential solutions for the inequality reside.
By picking a test point, such as \(0\), within this range, you can verify the solution set is graphed correctly, as the test point should satisfy the original inequality.
Open Circles
Open circles are essential when graphing inequalities that do not include the boundary values. They signify that a specific number is not part of the solution set. For the inequality \(-3 < x < 3\), place an open circle at \(-3\) and another at \(3\).
These circles highlight that while the number line includes values close to \(-3\) and \(3\), the numbers themselves are excluded. This distinction is important for accurately representing the inequality on a number line.
Always use open circles for inequalities with the symbols \(>\) or \(<\), and closed circles for \(\leq\) or \(\geq\) to show inclusion of boundary points.

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Most popular questions from this chapter

Consider the following statement: Today's high temperature will be at least 10 degrees lower than yesterday's high temperature. If the inequality \(T \leq t-10\) correctly represents this statement, what does the variable \(t\) represent?

A theorem from geometry called the Triangle Inequality Theorem states that the sum of the lengths of two sides of a triangle must be greater than the length of the third side. Suppose two sides of a triangle measure 10 in. and 18 in. Let \(x\) be the length of the third side. What are the possible values for \(x ?\)

Graph the set. (GRAPH CANNOT COPY) $$(-\infty, \infty)$$

A health official recommends a maximum cholesterol level of 200 units. By how many units must a patient with a cholesterol level of 275 units reduce her cholesterol level to satisfy the recommended maximum level?

Solve and write the answer in set-builder notation. $$d+\frac{1}{2}<\frac{1}{3}$$
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