Problem 5 Graph each inequality on a numbe... [FREE SOLUTION]

Chapter 2: Problem 5

Graph each inequality on a number line. Then write the solutions in interval notation. $$ x \leqq-1 $$

Short Answer

Expert verified

$( -\infty, -1 ]$

Step by step solution

Understanding the Inequality

First, let's interpret the inequality $ x \leq -1 $. This means that $ x $ can be any number less than or equal to $-1$. The values of $ x $ include all numbers to the left of $-1$ on the number line, including $-1$ itself.

Graph the Inequality

To graph $ x \leq -1 $ on a number line, draw a closed circle (or dot) on $-1$ to indicate that $-1$ is included in the solution. Then shade the number line to the left of $-1$ to show all the numbers that are less than $-1$.

Write in Interval Notation

In interval notation, the solution ensures that $ x $ includes $-1$ and all numbers less than $-1$. This is represented as $( -\infty, -1 ]$, where $-\infty$ is used to indicate that there is no lower bound and the $[\; ]$ bracket means $-1$ is included in the set.

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Key Concepts

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

graphical representation of inequalities

Graphical representation of inequalities is a powerful visual tool that allows us to easily interpret and understand the range of solutions for an inequality. When you graph an inequality like $ x \leq -1 $ on a number line, the first thing you need to do is identify the boundary point, which in this case is \,$-1$.

Here's how you do it:

Locate the number $-1$ on the number line.
Place a closed circle (filled dot) on $-1$ to indicate that this number is included in the solution set. The closed circle signifies that $-1$ is part of the solution because we are dealing with ''less than or equal to''.
Shade the entire part of the number line to the left of $-1$. This shaded region represents all the numbers that are less than $-1$.

The shaded area and the closed circle together denote that every number from negative infinity up to and including $-1$ is a solution to the inequality.

interval notation

Interval notation is a compact and efficient way to express a range of numbers, and it is especially useful for representing solutions to inequalities. In the inequality $ x \leq -1 $, we need to describe all the possible values of $ x $ that satisfy this condition.

Here's how interval notation works:

Starts with a parenthesis ''('' or a bracket ''['' and ends with a parenthesis '')'' or bracket '']''. Use a parenthesis to indicate that a number is not included (open interval) or a bracket when it is included (closed interval).
For $ x \leq -1 $, the interval notation is $(-\infty, -1]$. The parenthesis with $-\infty$ indicates the interval extends indefinitely to the left, meaning there is no minimum value. The bracket with $-1$ shows that this value is contained within the interval.

Understanding this notation helps in accurately expressing solution sets and in quickly interpreting mathematical conditions related to intervals.

solving linear inequalities

Solving linear inequalities is similar to solving equations, but there are a few extra rules to remember. When you solve an inequality, you're looking for the set of values that makes it true.

Here are the simple steps:

Identify the inequality you need to solve; for example, $ x \leq -1 $.
Think about what values of $ x $ make this inequality true. For example, any number that is $-1$ or less satisfies $ x \leq -1 $.
Solve the inequality as you would a simple equation, but keep in mind that if you multiply or divide by a negative number, the inequality sign flips direction. Luckily, we don't need to do that here.

The principal goal with solving inequalities is to isolate the variable on one side and then clearly identify the set of numbers (the domain) that satisfies the inequality condition. Mastering these steps makes it much easier to handle more complex inequalities moving forward.

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91影视

Graph each inequality on a number line. Then write the solutions in interval notation. $$ x \leqq-1 $$

Short Answer

Step by step solution

Understanding the Inequality

Graph the Inequality

Write in Interval Notation

Key Concepts

graphical representation of inequalities

interval notation

solving linear inequalities

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