/*! 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 41 Graph the given inequalities on ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 41

Graph the given inequalities on the number line. $$t \leq-5 \text { and } t \geq-5$$

Short Answer

Expert verified
The solution is \( t = -5 \); graph with a closed dot at \(-5\).

Step by step solution

01

Understand the Inequalities

The inequalities given are \( t \leq -5 \) and \( t \geq -5 \). This means that \( t \) is both less than or equal to \(-5\) and greater than or equal to \(-5\).
02

Determine the Solution Set

Since \( t \) must satisfy both conditions, it means \( t \) can only be \(-5\). This is because \(-5\) is the only value that is both less than or equal to and greater than or equal to \(-5\).
03

Graph the Solution on the Number Line

On the number line, represent \(-5\) with a closed dot or circle. The closed dot at \(-5\) indicates that \(-5\) is included in the solution set.

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
A number line is a straight, horizontal line used in mathematics to represent real numbers. Each point on this line corresponds to a real number, arranged in increasing order from left to right.
This visual tool helps in identifying the position of a number and the relationship between numbers, such as inequalities or ranges.
When representing numbers, particular attention is given to how they are marked to indicate if a specific number is part of the solution.
For example:
  • Open dots are used to indicate that a number is not included (used for strict inequalities like \( t < -5 \)).

  • Closed dots show that a number is included (for non-strict inequalities like \( t \leq -5 \)).
When graphing the inequality \( t \leq -5 \) and \( t \geq -5 \), a closed dot is placed on the number line at \(-5\), illustrating that \(-5\) is a solution.
Solution Set
The solution set of an inequality is the collection of numbers that fulfill the condition expressed by the inequality.
For inequalities like \( t \leq -5 \) and \( t \geq -5 \), you first determine where both conditions apply simultaneously.
  • This means finding the values of \( t \) that make both inequalities true.

  • In this case, the equality signs direct you to the border point \(-5\).
Because \(-5\) is both less than or equal to, and greater than or equal to itself, it is the only number that satisfies both conditions.
Thus, the solution set is a single element, \(-5\).
In problems involving solution sets, it's critical to determine where conditions overlap.
When multiple inequalities are involved, always check if there is overlap or if singular values satisfy every condition.
Mathematical Inequalities
Mathematical inequalities are expressions involving the symbols \( <, \leq, >, \text{and} \geq \) to demonstrate the relative size or order of two values.
They are used to show how one quantity is bigger, smaller, or at least as big or small as another.
For instance:
  • \( x < 3 \) denotes that \( x \) can be any number smaller than 3.

  • \( x \geq 3 \) implies \( x \) is 3 or greater.

  • Compound inequalities, like \( a \leq b \leq c \), indicate \( b \) equals or lies between \( a \) and \( c \).
In the provided exercise, the inequalities \( t \leq -5 \) and \( t \geq -5 \) show that the only possible value for \( t \) which satisfies both conditions simultaneously is \(-5\).
Understanding and manipulating inequalities is essential in solving problems with constraints and identifying possible solutions.

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

Give verbal statements equivalent to the given inequalities involving the number \(x\). $$-1 \leq x<3 \text { or } 5

Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. $$y \leq 0, x \geq 0, y \geq x$$

Graph the given inequalities on the number line. $$-4

Answer the given questions by solving the appropriate inequalities. The weight \(w\) (in \(\mathrm{N}\) ) of an object \(h\) meters above the surface of Earth is \(w=r^{2} w_{0} /(r+h)^{2},\) where \(r\) is the radius of Earth and \(w_{0}\) is the weight of the object at sea level. Given that \(r=6380 \mathrm{km}\), if an object weighs \(200 \mathrm{N}\) at sea level, for what altitudes is its weight less than 100 N?

Solve the given inequalities. Graph each solution. $$3 x+2 \leq 11$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.