Problem 3 - College Algebra

Question

In Exercises 1-12, graph the solutions of each inequality on a number line. $$x<-4$$

Step-by-Step Solution

Answer

The solution of the inequality \(x < -4\) includes all the numbers less than -4. This is graphically represented by an open circle at -4 on the number line, with a line or arrow extending towards the left demonstrating the values of x.

Step 1: Understand the inequality

The given inequality is \(x < -4\), where x represents an unknown value that is less than -4.

Step 2: Draw the number line

Draw a straight horizontal line to signify the number line. It is important that the line is long enough so that -4 and some values around it can be depicted.

Step 3: Mark -4 on the number line

Locate and mark -4 on the number line. Because the inequality is 'less than' and not 'less than or equal to', this number itself is not included into the solution. Consequently, a open circle (or parenthesis) is drawn above -4 to indicate this fact.

Step 4: Indicate the side of -4 where the solution is

Inequalities like \(x < -4\) signify that the solution is wherever x is less than -4. On the number line, less than -4 involves all the values to the left side of -4. Therefore, a line or arrow is drawn from the open circle at -4 and extended towards the left side to illustrate that the solution includes all the numbers less than -4.

Key Concepts

Number Line

A number line is a visual tool that helps us represent numbers in a straight line format. It is often used to make abstract mathematical concepts more tangible and easier to understand.

• The line is usually horizontal and extends infinitely in both directions.
• Numbers increase as we move right and decrease as we move left.
• Key numbers, such as integers, are marked by points on the line.

In the context of inequalities, the number line is crucial as it provides a clear picture of the range of possible solutions. When graphing an inequality like \(x < -4\), it shows what values x can take, making it easy to visualize the set of numbers that satisfy the condition.

Inequalities

Inequalities are mathematical statements that express the relative size or order of two values. An inequality shows how one quantity is different from another, often in terms of smaller or larger.
There are several types of inequalities:

• "Less than" (\(<\))
• "Less than or equal to" (\(\leq\))
• "Greater than" (\(>\))
• "Greater than or equal to" (\(\geq\))

In this exercise, we focused on \(x < -4\). This means x represents values that are strictly less than -4. Inequalities are essential in defining ranges of values, shaping the solutions evident on the number line.

Solution Set

The solution set of an inequality consists of all values that satisfy the inequality. It is crucial to understand that the solution set extends beyond just a single number.
For the inequality \(x < -4\):

• All numbers smaller than -4 are part of the solution set.
• Numbers like -5, -6, -7, etc., are included.
• The solution extends indefinitely towards negative infinity.

Depicting the solution set on a number line helps us see this infinite range of values visually. Arrows or lines directed from a marked point often illustrate this range to emphasize continuity.

Open Circle

An open circle on a number line is a marker used to indicate that a particular number is not included in the solution set.
When graphing the inequality \(x < -4\):

• The open circle is placed at -4.
• This shows that -4 itself is not part of the solution set.
• The open circle contrasts with a closed circle, which would mean inclusion in the solution set.

The open circle helps in clearly demonstrating the boundaries of the solution, ensuring there is no ambiguity about whether the boundary number satisfies the inequality or not.