Problem 51 $47-52$ : Express the inequali... [FREE SOLUTION]

Chapter 0: Problem 51

$47-52$ : Express the inequality in interval notation, and then graph the corresponding interval. $$ x>-1 $$

Short Answer

Expert verified

Interval: $(-1, \infty)$; Graph: open circle at $-1$ with arrow to the right.

Step by step solution

Understand the Inequality

The given inequality is $ x > -1 $. This means that $ x $ can be any number greater than $-1$. The inequality symbol '>' indicates that $-1$ is not included in the solution set.

Write in Interval Notation

Since $ x $ needs to be greater than $-1$, but not equal to it, we start the interval right after $-1$. The interval continues to infinity because there is no upper limit on $ x $. In interval notation, this is expressed as $(-1, \infty)$. Here, a parenthesis '(', indicates that $-1$ is not included, and an infinity symbol indicates an unbounded interval in the positive direction.

Graph the Interval

To graph $(-1, \infty)$, first draw a number line. Place a point at $-1$ and draw an open circle around it; an open circle indicates that $-1$ is not included in the interval. Draw a line or arrow extending to the right from $-1$ to indicate that all numbers greater than $-1$ are included. Keep extending the line or arrow indefinitely to the right, representing infinity.

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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 mathematical shorthand used to describe a range of numbers. It uses brackets or parentheses to indicate whether endpoints are included or excluded in the interval.

A parenthesis '(', ')' indicates that the endpoint is not included (also known as an open interval).
A bracket '[', ']' indicates that the endpoint is included (a closed interval).

Consider the inequality from the exercise: $ x > -1 $. In this case:

Since $ x $ is greater than $-1$ but does not include $-1$, we use a parenthesis: $(-1$.
There is no upper limit given, so $ x $ goes to infinity. We represent this with $ \infty $ and a parenthesis because infinity is not a number that can be included: $ ( \infty )$.

Putting it all together, the interval notation for $ x > -1 $ is $(-1, \infty)$. This communication is concise and clear, showing the full range of possible values for $ x $.

Graphing Inequalities

Graphing inequalities offers a visual representation of the solutions. It helps to understand the range of numbers satisfying the inequality. To graph the inequality expressed in interval notation $(-1, \infty)$, follow these steps:

Start with a number line, a horizontal line with marked numbers.
Locate $-1$ on the number line. Since $-1$ is not part of the solutions (indicated by $x > -1$), draw an open circle around it. This circle shows that the value at this point is excluded from the set of solutions.
From the open circle, draw a line extending to the right. An arrow or line to the right signifies that all numbers greater than $-1$ are included.
Extend the line or arrow indefinitely towards positive infinity, reinforcing that there is no upper boundary.

Graphing not only helps solidify understanding but also communicates the solution range in a visual, easily digestible format. This skill is crucial for interpreting inequalities effectively.

Number Line

A number line is a straight, horizontal line that visually represents numbers at equal intervals or distances apart. It's an essential tool in graphing inequalities, helping us display and understand where a particular set of numbers lies in relation to others.

The center of a number line is usually zero, with positive numbers on the right and negative numbers on the left.
Each point on the number line corresponds to a real number, providing a spatial idea of its value and relation.
In graphing inequalities like $ x > -1 $, using a number line becomes invaluable as it shows which numbers are part of the solution set.

When we place an open circle on $-1$ and extend the line infinitely to the right, we illustrate that the solution includes all numbers greater than $-1$. A number line supplies a straightforward and visual method to comprehend abstract mathematical concepts such as inequalities.

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91影视

\(47-52\) : Express the inequality in interval notation, and then graph the corresponding interval. $$ x>-1 $$

Short Answer

Step by step solution

Understand the Inequality

Write in Interval Notation

Graph the Interval

Key Concepts

Interval Notation

Graphing Inequalities

Number Line

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