Problem 55 Express the inequality in interv... [FREE SOLUTION]

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Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 55

Express the inequality in interval notation, and then graph the corresponding interval. $$x>-1$$

Short Answer

Expert verified

Interval notation: $(-1, \infty)$; graph uses an open circle at $-1$ and a ray to the right.

Step by step solution

Understand the Inequality

The given inequality is $x > -1$. This means that $x$ can take any value greater than $-1$, but not $-1$ itself.

Write the Inequality in Interval Notation

To express $x > -1$ in interval notation, we use an open interval because $-1$ itself is not included. Therefore, the interval notation is $(-1, \infty)$.

Graph the Interval on a Number Line

Draw a number line, and locate $-1$ on it. Use an open circle to indicate that $-1$ is not included in the interval. Draw a ray extending to the right from $-1$ to show that all numbers greater than $-1$ are included.

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Key Concepts

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

Interval Notation

Interval notation is a way to describe a set of numbers along a number line. It uses parentheses and brackets to show intervals of inclusivity and exclusivity. For the inequality $ x > -1 $, we write it as $(-1, \infty)$.
This means:

The parenthesis $(-1$ indicates that $-1$ is not included in the set of solutions.
$\infty)$ symbolizes that there is no upper limit to the values $x$ can take, extending to positive infinity.
Parentheses are always used with infinity because infinity is not a number, so it cannot be "included".

Interval notation is a succinct way of expressing ranges and is often used in higher mathematics because of its precision and clarity.

Number Line

A number line helps visualize intervals and inequalities. When representing inequalities like $x > -1$, we place the numbers on a horizontal line.
Here's how you do it:

First, identify the critical number in the inequality, which is $-1$ in this case.
Mark $-1$ on the number line.
Use an open circle on $-1$ because the inequality $x > -1$ means $-1$ itself is not included.
Shade or draw a ray to the right, starting from the open circle, indicating that all numbers greater than $-1$ are part of the solution.

The number line is a powerful tool for visually understanding the concept of inequalities and how they relate to intervals.

Inequality Graphing

Graphing an inequality provides a visual representation that is easy to understand. To graph $x > -1$ on a number line, first identify that $-1$ is where the inequality changes, but it is not part of the solution set.

Place an open circle on $-1$ to graphically show that it is excluded from the solution.
Draw an arrow extending to the right, starting from the open circle.
The arrow indicates that every number greater than $-1$ is included in the solution set, highlighting infinity.

This method provides a quick, clear way to communicate which numbers satisfy $x > -1$ and to convey the concepts behind the inequality visually.

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91影视

Express the inequality in interval notation, and then graph the corresponding interval. $$x>-1$$

Short Answer

Step by step solution

Understand the Inequality

Write the Inequality in Interval Notation

Graph the Interval on a Number Line

Key Concepts

Interval Notation

Number Line

Inequality Graphing

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