Problem 10 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$x \leq-3$$

Step-by-Step Solution

Answer

The graph is a number line with an arrow going left from the point x=-3, indicating all the points x such that \(x \leq -3\) are in the solution set. This also includes the point -3, which is shown with a filled-in circle.

Step 1: Understanding the inequality

Before we start to graph, it is important to understand what the inequality \(x \leq-3\) means. It includes all numbers x that are less than or equal to -3.

Step 2: Draw a number line

Draw a horizontal line with an arrow going to the right. This line will represent the number line.

Step 3: Marking the inequality symbol on the number line

Mark a filled-in circle at -3. The filled-in circle represents that -3 is included in the solution set (since the inequality sign is \(\leq\), meaning less than OR equal to), so we want to show this on the graph.

Step 4: Graphing the inequality

As the inequality \(x \leq-3\) represents all numbers less than or equal to -3, draw a line going from -3 to the left (this shows that the solution includes numbers less than -3). Note that the arrow should point to the left, indicating that all numbers less than -3 are part of the solution set.

Key Concepts

Number Line

A number line is a simple tool used in mathematics to represent numbers visually. It is typically a straight, horizontal line.
On this line, numbers are spaced evenly and can extend infinitely in both directions. The center point is usually 0, and numbers increase to the right and decrease to the left.

• Adding numbers is like moving to the right.
• Subtracting numbers is like moving to the left.

This visual approach helps in understanding mathematical concepts such as inequalities. It allows you to see where numbers are in relation to each other, making it easier to identify sets of numbers that satisfy certain conditions.

Inequality Symbols

Inequality symbols are used to express relationships between numbers that are not equal. The main inequality symbols are:

• < (less than)
• \( \leq \) (less than or equal to)
• > (greater than)
• \( \geq \) (greater than or equal to)

For example, in the inequality \(x \leq -3\), the symbol \( \leq \) indicates that \(x\) includes both numbers less than and equal to -3. This is important when graphing inequalities, as it helps determine how the solution set should be represented visually on a number line.

Solution Set

A solution set is the collection of values that satisfy an inequality or equation.
For the inequality \(x \leq -3\), the solution set includes all numbers that are less than or equal to -3. This set can be visualized on the number line to see which numbers make the inequality true.
When a solution set includes all numbers less than a specific value, it continues infinitely in the negative direction. This helps in understanding the full range of possible solutions the inequality includes.

Filled-in Circle

A filled-in circle on a number line is used to show that a particular number is part of the solution set.
In the inequality \(x \leq -3\), you'll place a filled-in circle at -3. This indicates that -3 is included as a solution because the inequality allows \(x\) to equal -3 as well as be less than -3.
When graphing, the filled-in circle helps visually communicate that the boundary number is included, differentiating it from open circles, which would indicate the opposite.