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Chapter 2: Problem 7

Graph all solutions on a number line and provide the corresponding interval notation. $$ -4

Short Answer

Expert verified
The solution is \((-4, \infty)\) and is graphed with an open circle on \(-4\) extending to the right.

Step by step solution

01

Understanding the Inequality

The inequality given is \(-4 < x\). This means that the variable \(x\) can take any value that is greater than \(-4\). It implies that \(x\) cannot be exactly \(-4\), but can be any value slightly more than \(-4\).
02

Graphing the Inequality on a Number Line

To graph this inequality, draw a number line and place an open circle at \(-4\) to show that \(x\) cannot be \(-4\). Then, shade the region to the right of \(-4\) to indicate that \(x\) includes all numbers greater than \(-4\).
03

Writing the Interval Notation

The interval notation for \(-4 < x\) is \((-4, \infty)\). The parenthesis \((\) around \(-4\) indicates that \(-4\) is not included in the set. The \(\infty\) symbol signifies that the interval extends indefinitely to the right, covering all larger numbers.

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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 representation of numbers in sequence on a straight line. It's a fundamental tool in mathematics for comparing and visualizing numbers. On the number line, each point corresponds to a real number, and the positions are equidistant. When dealing with inequalities, such as \(-4 < x\), the number line helps us visualize which numbers satisfy the condition.
  • Numbers to the left are smaller, and those to the right are larger.
  • It's marked in increasing order from left to right.
  • An open circle on the line (as seen at \(-4\) for \(-4 < x\)) indicates that a number is not included in the solution set.
By understanding a number line, we can accurately portray solutions for inequalities.
Graphing Inequalities on a Number Line
Graphing inequalities on a number line is a way to visually depict the solutions of an inequality. For the inequality \(-4 < x\), we begin by identifying the number \(-4\) on the number line.
Once found, draw an open circle around \(-4\). This circle shows that \(-4\) itself is not a solution, but numbers greater than \(-4\) are included.
The next step involves shading the area to the right of \(-4\) along the number line. The shaded region illustrates that \(-4 < x\) includes all numbers greater than \(-4\).
  • Use an open circle when the number is not part of the solution (e.g., less than or greater than).
  • If the number was included (e.g., less than or equal), a closed dot would be used instead.
This graphical approach helps in understanding which numbers satisfy the inequality conditions.
Expressing Inequalities in Interval Notation
Interval notation is a concise way to describe a range of numbers. It is particularly useful when dealing with inequalities. For the inequality \(-4 < x\), we express this range using interval notation as \((-4, \infty)\).
This notation has some important points to consider:
  • The round bracket \(\(\)\) next to \(-4\) indicates that \(-4\) is not included in the set.
  • The comma separates the starting point, \(-4\), from the end, \(+\infty\), which signifies the solutions continue indefinitely.
  • The infinity symbol \(\infty\) shows that the solutions go on without bound to the right.
Using interval notation is a simple and effective way to state all possible solutions for an inequality.

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Most popular questions from this chapter

Two brothers leave the house at the same time traveling in opposite directions. One averages 40 miles per hour and the other 36 miles per hour. How long does it take for the distance between them to reach 114 miles?

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 2 x-1<5 \text { and } 3 x-1<10 $$

An equilateral triangle with sides measuring 10 units is similar to another equilateral triangle with scale factor of 2: 3 . Find the perimeter of the unknown triangle.

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ x+7 \leq 5 \text { and } x-3 \geq-10 $$

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -15 \leq 23 x-15<45 $$
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