Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph the solutions of each inequality on a number line. $$-1

Step-by-Step Solution

Answer

The solution to the inequality is presented and confirmed on the number line. It contains all real numbers greater than -1 and less than 3, excluding -1 and 3 themselves.

Step 1: Identify the Inequalities

The compound inequality can be broken down into two parts. The first inequality is \(x > -1\) and the second inequality is \(x < 3\)

Step 2: Draw a Number Line

Draw a number line and mark the numbers -1 and 3. These are the boundary points of the solution set.

Step 3: Identify the Solution Set

Since the inequality is strictly 'greater than -1' and strictly 'less than 3', color the portion of the number line between -1 and 3, excluding -1 and 3. Use open circles at -1 and 3 to indicate these numbers are not themselves included in the solution.

Step 4: Test the Solution

Choose any number in the colored section of the line and see if it satisfies the original inequality. If it does, the solution is correct.

Key Concepts

Number Line

The number line is a straight, horizontal line that is used to represent numbers. It is often used in mathematics to graphically depict solutions to inequalities. Numbers on a number line increase from left to right, allowing us to easily see the order and size of numbers. For inequalities, certain methods are used to indicate the values included in a solution.

• Draw the number line, and include enough space to show relevant numbers clearly.
• Positions are marked to indicate specific numbers, relevant to the problem at hand.
• Open circles or closed circles are used at specific points to denote whether those values are included in or excluded from the solution set.

Using a number line simplifies visualizing and solving inequalities by providing a concrete representation.

Compound Inequality

A compound inequality involves two separate inequalities that are combined, usually using 'and' or 'or'. In the case of the original exercise, the compound inequality is expressed with 'and', signifying that both parts must be true simultaneously. The expression given is \[-1 < x < 3\]

• "-1 < x" means that x is greater than -1.
• "x < 3" means that x is less than 3.

The conjunction 'and' suggests that x needs to satisfy both these conditions at once. This results in a continuous range of numbers that fulfill the criteria of both inequalities together.

Solution Set

The solution set of an inequality includes all the possible numbers (or solutions) that satisfy the given inequality. For the compound inequality \(-1 < x < 3\), the solution set includes all real numbers that are greater than -1 but less than 3.When representing a solution set on a number line:- Use of open circles at the points that are not included (in this case, at -1 and 3) since the inequality is strict.- The space or line segment between these open circles represents all the numbers that belong to the solution set.Checking or marking segments assists in determining which values fit within the defined conditions of the inequality.

Strict Inequalities

Strict inequalities are types of inequalities where the solutions do not include the boundary points. The use of '<' or '>' indicates a strict inequality.For this problem, \[-1 < x < 3\]means that:- x is strictly greater than -1, signifying x cannot equal -1.- x is strictly less than 3, meaning x cannot equal 3.This is graphically represented on a number line through open circles at the points -1 and 3, which indicates these points are not part of the solution set. This careful differentiation is crucial to capturing the exact solution of a strict inequality.