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Chapter 8: Problem 51

Graph and write interval notation for each compound inequality. $$ x<-1 \text { or } x>4 $$

Short Answer

Expert verified
\((-\infty, -1) \cup (4, \infty)\)

Step by step solution

01

- Understand the compound inequality

The given compound inequality is composed of two separate inequalities connected by 'or'. It shows two separate ranges for the variable x: one where x is less than -1, and another where x is greater than 4.
02

- Identify the ranges

For the inequality segments: 1. For the inequality \(x < -1\), x can be any value less than -1.2. For the inequality \(x > 4\), x can be any value greater than 4.
03

- Write the inequality in interval notation

In interval notation, the two separate parts of the inequality are written as: 1. For \(x < -1\): \(-\infty, -1\) 2. For \(x > 4\): \(4, \infty\)Combining these using 'or' translates to \((-\infty, -1) \cup (4, \infty)\) in interval notation.
04

- Graph the inequalities

Plot the values on a number line: 1. Draw an open circle (indicating not inclusive) at -1, and shade the number line to the left to represent \((-fty, -1)\).2. Draw an open circle at 4, and shade the number line to the right to represent \((4, \infty)\).

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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 writing subsets of the real number line. It is a more compact alternative to conveying inequalities. The bounds of the interval are written as a pair of numbers inside parentheses or brackets. If the endpoint is included in the interval, we use a bracket [ ], otherwise, we use a parenthesis ( ).
For instance:
  • (-鈭�, -1) means all numbers less than -1.
  • [4, 鈭�) means all numbers greater than or equal to 4.

In the exercise provided, the interval notation for the compound inequality x < -1 or x > 4 was (-鈭�, -1) 鈭� (4, 鈭�), meaning all numbers less than -1 or greater than 4.
graphing inequalities
Graphing inequalities involves shading portions of the number line to show where the inequality holds true. For a given inequality:
  • Use an open circle to denote that a number is not included in the solution set (e.g., for < or >).
  • Use a closed circle to show that a number is included in the solution set (e.g., for 鈮� or 鈮�).

For example, for the inequality x < -1, draw an open circle at -1 and shade to the left. This shows all values less than -1. For x > 4, draw an open circle at 4 and shade to the right. This indicates all values greater than 4. Together, these create the visual representation of (-鈭�, -1) 鈭� (4, 鈭�).
number line representation
A number line is a visual representation of numbers in order. When representing inequalities on a number line, we use circles and shading to show ranges of solutions.
  • An open circle represents that the endpoint is not included in the interval.
  • An enclosed circle indicates inclusion of the endpoint.

For the inequality x < -1 or x > 4:
  • Label -1 and 4 on the number line.
  • Place an open circle at -1 and shade to the left.
  • Place an open circle at 4 and shade to the right.

    • This shading visually demonstrates that x is either less than -1 or greater than 4, matching the given inequality.

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Most popular questions from this chapter

Classify each of the following statements as either true or false. To solve \(\frac{x-5}{x+4} \geq 0\) using intervals, we divide the number line into the intervals \((-\infty,-4),(-4,5)\) and \((5, \infty)\).

For \(g(x)=(x-2)(x-3)(x+1),\) find all \(x\) -values for which \(g(x)>0\).

Solve. $$ 7.34 x^{2}-16.55 x-3.89 \geq 0 $$

Solve. $$ (x+5)(x+2)>0 $$

Find each solution set. $$ x^{4}+2 x^{2} \geq 0 $$
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