/*! 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 5 Find the graphical solution of e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 5

Find the graphical solution of each inequality. $$x \leq-3$$

Short Answer

Expert verified
The graphical solution of the inequality \(x \leq -3\) is a number line with a filled circle at -3 and shaded to the left, indicating the solution set includes all values less than or equal to -3.

Step by step solution

01

Draw a number line

Draw a horizontal number line, labeling key points, such as -5, -4, -3, -2, -1, 0, 1, etc.
02

Locate the value of x on the number line

Find the value \(x = -3\) on the number line and place a point on the number line at this point.
03

Distinguish the inclusion or exclusion of the point

Since the inequality is \(x \leq -3\), the solution set includes -3. Thus, draw a filled circle at the point \(x = -3\) on the number line.
04

Identify the direction of the inequality

The inequality states \(x \leq -3\), meaning the solution set consists of values less than or equal to -3. On the number line, values less than -3 are to the left of -3.
05

Shade the solution set

Shade the part of the number line that represents the solution set. In this case, shade to the left of the filled circle at -3, indicating that the solution set includes all values less than or equal to -3. The graphical solution of the inequality \(x \leq -3\) is a number line with a filled circle at -3 and shaded to the left.

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
Understanding the number line is essential for visualizing and solving mathematical problems, particularly inequalities. A number line is a straight, horizontal line that features points that correspond to numbers. The numbers are usually placed at equal intervals, and they increase in value as you move to the right, while decreasing as you move to the left. For instance, if we consider the inequality from the exercise, (\(x \leq-3\)), visualizing it on a number line can simplify the problem.

To do this, you start by drawing a line and marking it with numbers at regular intervals, ensuring to include the critical value from the inequality, which in our example is -3. The value of -3 on the number line serves as a reference point, from which we will determine the set of numbers that are solutions to the inequality. Number line representations offer a comprehensible way to look at the range of possible solutions for inequalities. The graphical solution, with a filled circle at -3, represents that -3 is included in the solution set. The shaded part of the number line indicates every number less than -3 also belongs to the solution set.
Solving Inequalities
Inequalities are mathematical expressions indicating that two values are not equal, in contrast to equations that convey exact equality. Solving inequalities involves finding all the possible values for the variables that would make the inequality true. The steps for this are systematic and are key to understanding the types of solutions that are acceptable.

When facing a problem such as (\(x \leq -3\)), the process begins with identifying the relational symbol. In this case, it's (\leq), which means 'less than or equal to'. Next, you imagine the set of all numbers that satisfy this condition. The solution includes not just a single number, but a range of numbers. It's crucial to determine whether the number associated with the variable, -3 here, should be included or not. The (\leq) symbol means that -3 is indeed a part of the solution, which is depicted graphically as a filled circle on the number line. Communicating the solution to an inequality requires showing this range properly, which brings us to the visual aspect of these tasks鈥攖he graphical representation.
Graphical Representation of Mathematical Concepts
Graphical tools are extremely useful in mathematics for conveying information that can be hard to grasp through numbers alone. The graphical representation of mathematical concepts, such as inequalities, allows students to visualize the set of possible solutions in a clear and intuitive manner.

When you graph (\(x \leq -3\)) on a number line, the visual impact is immediately apparent. A student can see that every point to the left of -3 is included, due to the direction in which we shade the number line. This visual aid can be more effective for understanding the concept than a verbal or written explanation. Graphical representations help bridge the gap between abstract numerical expressions and their practical interpretations.

In teaching, using a graphical approach enables students to not only see the solutions but to understand the logic behind them. This transition from numerical to visual understanding is a significant step in building a deeper comprehension of mathematics.

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

The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gal of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $$\$ 300$$ and that for pipeline water is $$\$ 500$$, how much water should the manager get from each source to minimize daily water costs for the city? What is the minimum daily cost?

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 60 deluxe cabins and 160 standard cabins. whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 -day cruise in May. It costs $$\$ 44,000$$ to operate a type-A vessel and $$\$ 54,000$$ to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. She has decided that the meals should contain a minimum of \(400 \mathrm{mg}\) of calcium, \(10 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin C. She has further decided that the meals are to be prepared from foods \(\mathrm{A}\) and \(\mathrm{B}\). Each ounce of food \(\mathrm{A}\) contains \(30 \mathrm{mg}\) of calcium, \(1 \mathrm{mg}\) of iron, \(2 \mathrm{mg}\) of vitamin \(\mathrm{C}\), and \(2 \mathrm{mg}\) of cholesterol. Each ounce of food \(\mathrm{B}\) contains \(25 \mathrm{mg}\) of calcium, \(0.5 \mathrm{mg}\) of iron, \(5 \mathrm{mg}\) of vitamin \(\mathrm{C}\), and \(5 \mathrm{mg}\) of cholesterol. Find how many ounces of each type of food should be used in a meal so that the cholesterol content is minimized and the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met.

Custom Office Furniture is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and \(\mathrm{C}\) -are to be marketed. Each model-A desk requires \(1 \frac{1}{4} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each model-B desk requires \(1 \frac{1}{2} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each modelC desk requires \(1 \frac{1}{2} \mathrm{hr}, \frac{3}{4} \mathrm{hr}\), and \(\frac{1}{2} \mathrm{hr}\) for fabrication, assembly, and finishing, respectively. The profit on each model-A desk is $$\$ 26$$, the profit on each model-B desk is $$\$ 28$$, and the profit on each model-C desk is $$\$ 24$$. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is \(310 \mathrm{hr}, 205 \mathrm{hr}\), and \(190 \mathrm{hr}\), respectively. To maximize Custom's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.