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Chapter 0: Problem 127

What is a compound inequality and how is it solved?

Short Answer

Expert verified
A compound inequality is a statement with two inequalities combined by 'and' or 'or'. An 'and' compound inequality requires both conditions to be satisfied, resulting in a solution set that satisfies both inequalities. An 'or' compound inequality requires that at least one of the conditions is satisfied, resulting in a solution that satisfies either of the inequalities. The compound inequalities can be represented on a number line with a segment for 'and' and two rays for 'or'.

Step by step solution

01

Definition of a Compound Inequality

A compound inequality is a pair of inequalities which are combined into one statement by the words 'and' or 'or'. When two inequalities are connected by the word 'and', we are looking for the set of numbers that satisfy both inequalities. When two inequalities are connected by the word 'or', we are looking for the set of numbers that satisfy either of the inequalities.
02

Understanding 'and' Compound Inequalities

For an 'and' compound inequality, both conditions in the inequalities must be met. For example, the compound inequality \(2 < x \, \text{and} \, x \, < 5\) means that \(x\) is greater than 2 and less than 5, represented as \(2 < x < 5\). To solve, we need to find the values of \(x\) which satisfy both conditions.
03

Understanding 'or' Compound Inequalities

For an 'or' compound inequality, either one or both of the conditions in the inequalities must be met. For example, the compound inequality \(x < 2 \, \text{or} \, x > 5\) means that \(x\) is either less than 2 or greater than 5. To solve, we need to find the values of \(x\) which satisfy at least one condition.
04

Representation on Number Line

Compound inequalities can be represented on a number line. For an 'and' compound inequality, we will have a closed segment on the number line. For an 'or' compound inequality, we will have two rays extending in opposite directions on the number line.

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical expressions that compare two values or expressions using signs like \(<\), \(\leq\), \(>\), or \(\geq\). They show the relationship between the values, indicating if one is greater, equal, or less than the other.
  • A simple inequality might look like \(x < 3\), meaning \(x\) can be any number less than 3.
  • Inequalities are similar to equations but instead of equality, they represent a range of possible solutions.
Understanding inequalities is essential for solving complex problems like compound inequalities, where multiple inequalities are bundled together.
Using a Number Line
A number line is a visual tool that helps us understand the solutions of inequalities and compound inequalities. It鈥檚 a straight line where numbers are ordered sequentially.
  • For instance, if we have the inequality \(x > 4\), we would express this on a number line by shading all numbers greater than 4.
  • Number lines help in quickly visualizing the range of solutions. They can show both separate solutions and overlapping solutions for compound inequalities.
In the case of \(2 < x < 5\), the number line would be shaded between 2 and 5, not including the endpoints.
Mathematical Representation of Compound Inequalities
A compound inequality is a combination of two or more inequalities joined by 'and' or 'or'. They are written in forms like \(a < x < b\) for 'and', or \(x < a \) or \( x > b\) for 'or'.
  • 'And' means both conditions must be true, creating a range of solutions.
  • 'Or' means any condition can be true, often representing solutions on opposite sides of a number line.
Representing these inequalities mathematically helps in visualizing and solving them systematically.
Inequality Solution Methods
Solving compound inequalities requires understanding how to address both 'and' and 'or' connections.
  • 'And' inequalities (e.g., \(2 < x < 5\)) are solved by finding a common range satisfying both inequalities. Here, the solution is simply all values of \(x\) between 2 and 5.
  • 'Or' inequalities (e.g., \(x < 2\) or \(x > 5\)) allow solutions from either inequality. This often results in two separate ranges displayed on a number line.
It's about identifying which parts of the inequalities overlap or diverge. This approach helps in reaching logical conclusions about the possible values of the variable.

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