Problem 5 Graph all solutions on a number ... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 2: Problem 5

Graph all solutions on a number line and provide the corresponding interval notation. $$ x \leq-3 $$

Short Answer

Expert verified

The interval is $(-\infty, -3]$, represented by shading left of $-3$ on the number line with a solid dot at $-3$.

Step by step solution

Understand the Inequality

The inequality given is $ x \leq -3 $. This means that $ x $ can be any number that is less than or equal to $-3$.

Determine the Interval

Since $ x $ can be any number less than or equal to $-3$, the interval in interval notation is $(-\infty, -3]$. The parenthesis $(-\infty$ indicates that negative infinity is not included, and the bracket $[-3]$ means $-3$ is included.

Plot on the Number Line

Draw a horizontal line representing the number line. Locate $-3$ and draw a solid dot on it to indicate $-3$ is included in the solution set. Shade the entire number line to the left of $-3$ to represent all values less than $-3$.

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Key Concepts

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

Number Line

A number line is a simple visual representation that helps us see the position of numbers. It's a straight horizontal line with numbers placed at equal intervals along it. In working with inequalities, a number line is used to show the set of possible solutions.
For the inequality $ x \leq -3 $, the number line helps us graphically represent all the values that $ x $ can take. We use a solid dot to indicate $-3$, as $ x $ can equal $-3$. Then, we shade the line to the left of $-3$.

The use of a solid dot confirms that $-3$ itself is part of the solution set.
Shading to the left shows that all values less than $-3$ are included.

This visual method is powerful, as it provides an immediate understanding of the inequality's solutions.

Interval Notation

Interval notation is a mathematical notation used to describe a set of numbers on the number line. It is concise and unambiguous, making it useful for conveying the range of solutions in inequalities.
For $ x \leq -3 $, interval notation is written as $(-\infty, -3]$. Here's how it works:

The parenthesis $(-\infty$ indicates that negative infinity is not included. In mathematics, infinity is a concept rather than a number, so it can't be reached or included.
The bracket $[-3]$ means that $-3$ is included in the interval. When you see a bracket, it tells you that the endpoint is part of the solution set.

This notation allows us to communicate the idea of extending a solution indefinitely in one or both directions.

Graphing Inequalities

Graphing inequalities involves plotting the solutions of an inequality on a number line or coordinate plane. It provides a visual way to understand which numbers satisfy the condition given by the inequality.
To graph $ x \leq -3 $, follow these steps on a number line:

Mark $-3$ with a solid dot because the inequality includes equality ($ x = -3 $ is valid).
Shade all points to the left of $-3$ on the number line. This shading vividly shows us that all these points are valid solutions for $ x $.
The choice of a dashed or solid line will depend on whether the inequality is strict ($<$, $>$) or includes equality ($\leq$, $\geq$). Solid lines or dots are used when the number at the endpoint is part of the solution.

Graphing is a valuable skill that enhances comprehension and allows for quick identification of solutions in inequalities.

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91影视

Graph all solutions on a number line and provide the corresponding interval notation. $$ x \leq-3 $$

Short Answer

Step by step solution

Understand the Inequality

Determine the Interval

Plot on the Number Line

Key Concepts

Number Line

Interval Notation

Graphing Inequalities

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