/*! 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 6 Solve each linear inequality and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 6

Solve each linear inequality and graph the solution set on a number line. $$-5 x \leq 30$$

Short Answer

Expert verified
The solution to the inequality is \(x \geq -6\).

Step by step solution

01

Isolate the variable term

Start by isolating x. Since -5 is multiplied with x, start by dividing both sides by -5 to get x on its own. Remember to flip the inequality sign when dividing by a negative number: \(x \geq -6\)
02

Graphing the solution set

To graph this solution set, a number line is needed. Put -6 in the middle and put a closed circle at -6 because the inequality is 'greater than or equal to'. Draw a line to the right of -6 to represent 'all numbers greater than -6'.

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.

Graphing Inequalities
Graphing inequalities allows us to visualize solutions on a number line and understand the range of valid solutions. When graphing an inequality, it is important to pay attention to whether it involves a strict inequality (like `<` or `>`) or an inclusive one (like `鈮 or `鈮). This distinction affects how we represent the solution on the number line.

  • Strict Inequality (< or >): Use an open circle on the number line to indicate the specific point is not included in the solution set.
  • Inclusive Inequality (≤ or ≥): Use a closed circle to show the point is included.
For example, the inequality \(x \geq -6\) requires placing a closed circle at -6. We then shade or draw a line extending to the right, indicating all numbers greater than -6 are solutions.
Solving Inequalities
Solving inequalities is very similar to solving equations, but there is one critical difference: reversing the inequality sign. This happens when both sides of the inequality are multiplied or divided by a negative number.

In the inequality \(-5x \leq 30\):
  • We divide both sides by -5 to isolate x.
  • Because we're dividing by a negative number, we reverse the inequality sign, changing \(\leq\) to \(\geq\).
  • This results in the new inequality: \(x \geq -6\).
This rule is essential because it maintains the truth of the inequality. In essence, it keeps the relationship between both sides unchanged when multiplied or divided by negative numbers.
Number Line Graph
A number line graph provides a simple yet powerful visual tool for representing solutions to inequalities. It enables quick visual inspection to determine the solution set at a glance. Here鈥檚 how to effectively use a number line graph to display solutions:

  • Marking Points: Identify the key points from the solution. For example, in \(x \geq -6\), point -6 is crucial.
  • Using Circles: Use circles at your points. Closed circles for \(\leq\) or \(\geq\), and open circles for \(\lt\) or \(\gt\).
  • Shading Regions: Shade or draw a line outward from the circle to show all possible solutions. To the right for inequalities like \(x \geq -6\).
Using these techniques provides clarity and precision in showcasing the range of solutions, helping you understand more about the nature 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

On the first four exams, your grades are \(70,75,87,\) and 92. There is still one more exam, and you are hoping to earn a \(\mathrm{B}\) in the course. This will occur if the average of your five exam grades is greater than or equal to 80 and less than \(90 .\) What range of grades on the fifth exam will result in earning a B? Use interval notation to express this range.

Write each sentence as a linear inequality in two variables. Then graph the inequality. The \(y\) -variable is at least 2 more than the product of \(-3\) and the \(x\) -variable.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The smallest real number in the solution set of \(2 x>6\) is 4.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Compound inequalities with and have solutions that satisfy both inequalities, whereas compound inequalities with or have solutions that satisfy at least one of the inequalities.

Will enable you to review graphing linear functions. In addition, they will help you prepare for the material covered in the next section. In each exercise, graph the linear function. \(3 x-5 y=15\) (Section 3.2, Example 5)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.