/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Express the inequality interval ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 55

Express the inequality interval notation, and then graph the corresponding interval. $$-2

Short Answer

Expert verified
The interval notation is \\((-2, 1]\\); graph with an open circle at \\(-2\\) and a closed circle at \\(1\\).

Step by step solution

01

Identify the type of inequality

Examine the inequality \((-2 < x \leq 1)\). This is a compound inequality because it has two parts: \((-2 < x)\) and \(x \leq 1)\). This indicates that the interval extends from \(-2\) to \(1\).
02

Determine Inclusion or Exclusion of Endpoints

For the portion \((-2 < x)\), the interval does not include \(-2\), since it is a strict inequality. For \(x \leq 1)\), the interval includes \(1\) because it is a \(\leq\) inequality, which means \(x\) can be equal to \(1\).
03

Write Interval Notation

Using findings from Step 2, translate the inequality to interval notation. Use a parenthesis \(\bigl(\bigr)\) to indicate a number is not included and a square bracket \[\bigl[\bigr]\] to indicate that it is included. Therefore, the interval notation is \((-2, 1]\).
04

Graph the Interval

To graph \((-2, 1]\), place an open circle at \(-2\) to indicate that \(-2\) is not included and a closed circle at \(1\) to show that \(1\) is included. Draw a line connecting these two points to represent all numbers \(x\) that satisfy the inequality.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Compound Inequality
A compound inequality typically involves two separate inequalities joined by the word "and" or "or". In our given exercise, the compound inequality is \(-2 < x \leq 1\), meaning we look for values of \(x\) that satisfy both \(-2 < x\) and \(x \leq 1\).

Understanding compound inequalities is important because they define a range of values that a variable can take.
Consider the following:
  • The statement \(-2 < x\) means \(x\) can be any number greater than \(-2\) but not equal to \(-2\).
  • On the other side, \(x \leq 1\) means \(x\) can be less than or equal to 1.
Compound inequalities can appear in various real-world situations. For example, they can represent temperature ranges, budget constraints, or limits in scientific measurements. Understanding how to work with them provides important problem-solving skills.
Interval Notation
Interval notation is a concise way of writing subsets of real numbers, which is particularly useful for expressing solutions to inequalities. When writing in interval notation:
  • We use a parenthesis \((\) or \()\) to show that a number at the end of an interval is not included.
  • We use square brackets \([\) or \(]\) to indicate that a number at the boundary of an interval is included.

In the exercise \(-2 < x \leq 1\), converting this into interval notation involves:- Not including \(-2\), so it's noted as \((-2\).- Including \(1\), so it's expressed as \(1]\).

Thus, the interval notation for this inequality is \((-2, 1]\). This method of notation allows us to quickly understand and view the span of possible solutions for a given inequality.
Graphing Inequalities
Graphing inequalities helps visualize the range of solutions. When graphing an inequality on a number line, communicate which numbers are included in the solution set:
  • An open circle is used to represent a number that is not part of the inequality solution.
  • A closed circle indicates that a particular number is included as part of the solution.

For \((-2 < x \leq 1\):
- Place an open circle at \(-2\) to indicate that it's not part of the solution.
- Place a closed circle at \(1\) because \(1\) is included in the solution.
- Connect these points by drawing a line between them to show all the numbers in between satisfy the inequality.
Graphing inequalities not only makes understanding solutions easier but also helps when solving systems or determining intersections of inequalities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Stopping Distance For a certain model of car the distance \(d\) required to stop the vehicle if it is traveling at \(v \mathrm{mi} / \mathrm{h}\) is given by the formula $$ d=v+\frac{v^{2}}{20} $$ where \(d\) is measured in feet. Kerry wants her stopping distance not to exceed 240 ft. At what range of speeds can she travel? (image cannot copy)

Simplify the expression. (a) \(\left(\frac{x^{3 / 2}}{y^{-1 / 2}}\right)^{4}\left(\frac{x^{-2}}{y^{3}}\right)\) (b) \(\left(\frac{4 y^{3} z^{2 / 3}}{x^{1 / 2}}\right)^{2}\left(\frac{x^{-3} y^{6}}{8 z^{4}}\right)^{1 / 3}\)

Rationalize Put each fractional expression into standard form by rationalizing the denominator. (a) \(\frac{12}{\sqrt{3}}\) (b) \(\sqrt{\frac{12}{5}}\) (c) \(\frac{8}{\sqrt[3]{5^{2}}}\)

Show that the equation represents a circle, and find the center and radius of the circle. $$3 x^{2}+3 y^{2}+6 x-y=0$$

Prove the following Laws of Exponents. (a) Law \(6:\left(\frac{a}{b}\right)^{-n}=\frac{b^{n}}{a^{n}}\) (b) Law \(7: \frac{a^{-n}}{b^{-m}}=\frac{b^{m}}{a^{n}}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.