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Chapter 8: Problem 52

Graph and write interval notation for each compound inequality. $$ x<-5 \text { or } x>1 $$

Short Answer

Expert verified
In interval notation: \((-umpy.\, -5) \cup (1, \infty)\).

Step by step solution

01

Understand the Inequalities

The given compound inequality is: 1. \(x < -5\) 2. \(x > 1\). This means the solution includes all values of \(x\) that are either less than -5 or greater than 1.
02

Represent Each Inequality on a Number Line

Draw a number line. Mark the point \(-5\) on the number line and shade all values to the left of \(-5\). This represents the solution to \(x < -5\). Then, mark the point 1 on the number line and shade all values to the right of 1. This represents the solution to \(x > 1\).
03

Write the Interval Notation

In interval notation, \(x < -5\) is written as \((-\infty, -5)\) and \(x > 1\) is written as \((1, \infty)\). Since the compound inequality uses 'or', combine these intervals: \((-\uE000,\textup{infty}, -5) \cup (1, \infty)\).
04

Combine the Graph

Combine the shaded regions from both inequalities on a single number line. Use open circles to indicate that -5 and 1 are not included in the solution.

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Key Concepts

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

interval notation
Interval notation is a way of representing a range of numbers, often used for inequalities. It condenses the solution set into a compact form, making it easy to read.

For example, the inequality \(x < -5\) is expressed in interval notation as \((-\infty, -5)\). Here:
  • \(-\infty\) represents negative infinity, indicating that there is no lower bound on our solutions.
  • The round bracket \((-5)\) shows that -5 itself is not included in the solution.

Similarly, for the inequality \(x > 1\), the interval notation is \((1, \infty)\). This means any number greater than 1 works, while 1 itself is excluded.
Combining these, because we are dealing with 'or', we write \((-\infty, -5) \cup (1, \infty)\). The \(\cup\) denotes the union of the two intervals.
number line
A number line is a helpful visual tool for representing solutions to inequalities. It allows you to see at a glance which ranges of values are included or excluded.
In the given problem, we have two separate inequalities: \(x < -5\) and \(x > 1\).

To graph these:
  • Draw a horizontal line and mark points -5 and 1 on it.
  • For the inequality \(x < -5\), shade the region to the left of -5. This indicates all values less than -5.
  • For the inequality \(x > 1\), shade the region to the right of 1, indicating all values greater than 1.

Remember to use open circles at -5 and 1 to show that these points are not included in the solution.
graphing inequalities
Graphing inequalities on a number line helps you visualize the solution set effectively.
In our example, we are graphing two inequalities: \(x < -5\) and \(x > 1\). When graphing:
  • Start by drawing the number line and marking the significant points, -5 and 1.
  • For \(x < -5\): Shade everything to the left of -5. Use an open circle at -5 to show it is not included.
  • For \(x > 1\): Shade everything to the right of 1. Use an open circle at 1 to show it's not included.

Once both parts are shaded, you have a combined graph illustrating that \(x\) can be any number less than -5 or greater than 1.
This combined graph visually represents the interval notation \-\((-\infty, -5)\)\cup\((1, \infty)\).

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