/*! 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 40 graph the given inequalities on ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 40

graph the given inequalities on the number line. \(x \leq 4\) or \(x > -4\)

Short Answer

Expert verified
Shade everything left of 4 including 4, and everything right of -4 on the number line.

Step by step solution

01

Identify the Inequalities

We have two inequalities to work with: one is \(x \leq 4\) and the other is \(x > -4\). These inequalities will be represented on the number line.
02

Graph the Inequality \(x \leq 4\)

To represent \(x \leq 4\), draw a solid dot (indicating 'equal to') on the number 4 on the number line and shade all the numbers to the left, including 4.
03

Graph the Inequality \(x > -4\)

For \(x > -4\), place an open dot (indicating 'not equal to') on the number -4 and shade everything to the right, excluding -4 itself.
04

Combine the Inequalities on the Number Line

Combine both graphs on a single number line. The region that satisfies at least one of the inequalities will be shaded. Since it's an 'or' statement, a point on the number line only needs to satisfy one condition.

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.

Number Line Representation
A number line is a visual representation of numbers on a straight, horizontal line. It's useful for displaying solutions to inequalities.
In the context of inequalities like \(x \leq 4\) or \(x > -4\), a number line helps visually show which values of \(x\) satisfy these conditions.
  • For \(x \leq 4\), a solid dot at 4 indicates that "4" is included.
  • Shade the line to the left of 4 to include all numbers less than or equal to it.
  • For \(x > -4\), use an open dot at -4 because "-4" is not part of the solution.
  • Shade all numbers greater than -4 to the right.
When multiple inequalities are combined, as here, all applicable regions are illustrated on the same number line, showcasing areas where one or more conditions are met.
Inequality Graphing
Graphing inequalities on a number line involves visually representing mathematical statements where one quantity is larger or smaller than another.
Starting with a basic understanding:
  • \(x \leq 4\): includes 4 and all numbers to its left; graphed with a solid dot at 4 and a shaded line extending left.
  • \(x > -4\): doesn't include -4 itself; graphed with an open dot at -4 and a shaded line extending to the right.
To graph, identify whether it's a "less than or equal to" or "greater than" type, and use open or solid dots accordingly.
Combine multiple inequalities by shading areas where any conditions are satisfied.
The result is a clear, combined illustration on the line.
Mathematical Symbols in Inequalities
Mathematical symbols are crucial for understanding and expressing inequalities accurately.
They define the relationship between quantities and guide how we represent them on a graph.
Key symbols include:
  • "\(\leq\)" means "less than or equal to" and indicates inclusion of the boundary number, shown with a solid dot.
  • "\(>\)" means "greater than" and excludes the boundary number, represented with an open dot.
These symbols clarify whether a border point is included in a solution set.
By using these notations, we can easily translate mathematical statements into a visual format on a number line.
Understanding such symbols is vital for correctly graphing and interpreting 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

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. Determine the values of \(x\) that are in the domain of the function \(f(x)=1 / \sqrt{3-0.5 x}\).

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. One pump can remove wastewater at the rate of 75 gal/min, and a second pump works at the rate of \(45 \mathrm{gal} / \mathrm{min} .\) Graph the possible values of the time (in min) that each of these pumps operates such that together they pump more than 4500 gal.

Use inequalities to solve the given problems. For what values of real numbers \(a\) and \(b\) does the inequality \((x-a)(x-b)<0\) have real solutions?

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. Determine the values of \(x\) that are in the domain of the function \(f(x)=\sqrt{2 x-10}\).

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. The voltage drop \(V\) across a resistor is the product of the current \(i\) (in \(\mathrm{A}\) ) and the resistance \(R\) (in \(\Omega\) ). Find the possible voltage drops across a variable resistor \(R\), if the minimum and maximum resistances are \(1.6 \mathrm{k} \Omega\) and \(3.6 \mathrm{k} \Omega,\) respectively, and the current is constant at \(2.5 \mathrm{mA}\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.