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Chapter 2: Problem 31

For Problems \(19-34\), graph the solution set for each compound inequality, and express the solution sets in interval notation. $$ x>-2 \text { or } x<3 $$

Short Answer

Expert verified
The solution set is \((-2, 3)\) in interval notation.

Step by step solution

01

Understand Each Inequality Separately

First, understand the two parts of the compound inequality separately. The first inequality is \( x > -2 \), which means \( x \) includes all values greater than \(-2\). The second inequality is \( x < 3 \), which represents all values less than \(3\).
02

Identify the Type of Compound Inequality

Since the compound inequality uses "or," it means we are looking for values of \( x \) that satisfy either one of the inequalities or both. This type of inequality results in a solution set that includes all values that satisfy at least one of the inequalities.
03

Combine the Inequalities and Graph

On a number line, plot both inequalities: \( x > -2 \) is represented by an open circle at \(-2\) and a line extending to the right. \( x < 3 \) is represented by an open circle at \(3\) and a line extending to the left. Since it's an "or" compound inequality, combine these, resulting in a graph that shows all numbers from \(-2\) to \(3\), not including \(-2\) and \(3\) themselves.
04

Express in Interval Notation

The entire solution set on the number line includes every number greater than \(-2\) or less than \(3\). In interval notation, this is expressed as: \((-2, 3)\). This notation signifies that values are included from just above \(-2\) up to just below \(3\), not including the endpoints.

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Key Concepts

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

Graphing Inequalities
When working with inequalities, visualizing them on a number line is a powerful way to understand solutions. Let's explore how this works with the compound inequality \(x > -2 \text{ or } x < 3\).
To graph this inequality, start by focusing on each part separately.
  • The inequality \(x > -2\) indicates that we need to plot all values greater than -2 on the number line. A handy way to represent this is to draw an open circle (which means -2 is not included) at -2 and shade the line extending to the right.
  • Similarly, for \(x < 3\), place an open circle at 3 and shade the line extending to the left, representing all numbers less than 3.
For compound inequalities involving "or," you combine both graphs. So for any \(x\), if it satisfies either \(x > -2\) or \(x < 3\), it's part of the solution. Thus, on the number line, you'd have a section shaded for all values from -2 to 3 except -2 and 3 themselves.
Interval Notation
Interval notation is a concise, mathematical language to describe the range of solutions for inequalities. It provides a clear visual of which numbers are included in a set.For the inequality \(x > -2 \text{ or } x < 3\), the interval notation is \((-2, 3)\). Here's why:
  • The use of parentheses \((\) and \()\) denotes that endpoint values are not included. Thus, -2 and 3 are not part of the solution, aligning with the open circles on the graph.
  • If the solution included the exact boundaries, we'd use brackets \([\) and \()]\) instead, showing that those values are part of the solution.
  • Interval notation is read from left to right, showing the span from just above -2 to just below 3 for our example.
Utilizing interval notation simplifies complex statements and conveys precise numerical ranges effectively.
Inequality Solutions
Solving inequalities provides insights into the range of possible values a variable can take. In our compound inequality \(x > -2 \text{ or } x < 3\), interpreting this solution means understanding which values satisfy the inequality.For compound inequalities:
  • "Or" statements are satisfied if any part is true. Any number greater than -2 or less than 3 fits the solution.
  • There鈥檚 no "intersection"; the solution doesn't need to satisfy both parts simultaneously, unlike "and" compound inequalities.
  • Check solutions by substituting numbers back into the original inequalities to ensure they maintain their truth in that range.
The flexibility of "or" inequalities creates broader solutions, capturing a wide span of numbers on the number line. This understanding reinforces how inequalities get solved and checked, reflecting different practical scenarios across a range of applications.

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Most popular questions from this chapter

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ 1<4 x+3<9 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -7<3 x-1<8 $$

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 3 x+2<-1 \text { or } 3 x+2>1 $$

Solve each inequality. $$ |-3 x-4| \leq 15 $$

For Problems \(1-16\), solve each equation. $$ |4-2 x|=6 $$
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