/*! 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 24 Solve each inequality. Graph the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 24

Solve each inequality. Graph the solution set. $$ -5 x<20 $$

Short Answer

Expert verified
The solution is \(x > -4\) and it is graphed as an open circle at \(-4\) extending right.

Step by step solution

01

Isolate the Variable

To solve the inequality \(-5x < 20\), we need to isolate \(x\) on one side. We can do this by dividing each side of the inequality by \(-5\). Remember, when dividing an inequality by a negative number, we must flip the inequality sign. Divide both sides to get: \(x > -4\).
02

Graph the Solution Set

On a number line, represent the solution \(x > -4\). Place an open circle at \(-4\) to indicate that \(-4\) is not included in the solution. Draw a ray extending to the right from \(-4\) to represent all numbers greater than \(-4\).

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.

Isolating Variables
Isolating the variable is a crucial step when solving inequalities because it helps to find what values a variable can take. Let's take a closer look at how to do this efficiently.To isolate a variable like in the inequality \(-5x < 20\), our main goal is to have the variable \(x\) on one side by itself. Here鈥檚 how you can achieve it:
  • First, consider the operation currently performed on \(x\), which in this case is multiplication by \(-5\).
  • To undo this, perform the opposite operation. Divide both sides of the inequality by \(-5\).
  • It's important to remember a critical rule: Whenever you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. This means \(<\) becomes \(>\).
Executing these steps, we have \(x > -4\).By isolating \(x\), you get a clearer picture of the range \(x\) belongs to and prepare yourself for graphing the solutions.
Graphing Solutions
Graphing the solutions of an inequality gives a visual representation of all the possible values the variable can take. Let鈥檚 learn how to graph the solution \(x > -4\) on a number line effectively.
  • Start by drawing a straight horizontal line, which will serve as your number line.
  • Identify the critical point from the solution, which is \(-4\) in this scenario.
  • Because the inequality is a strict \(>\) (greater than but not equal to), indicate this by placing an open circle on the number line at \(-4\). The open circle shows that \(-4\) is not included in the solution.
  • Next, draw a ray (a line with an arrow) starting at \(-4\) and extending to the right. This represents all numbers greater than \(-4\).
This simple graphical method helps clearly indicate the solution set on a number line, making it easier to interpret the inequality's results.
Inequality Rules
Understanding inequality rules is vital when solving them, as they guide how operations affect the inequality sign and ultimately the solution.Here are some essential rules to always remember:
  • **Transposing terms and maintaining the normal order:** If you add or subtract a term from both sides, the direction of the inequality remains unchanged.
  • **Multiplying or dividing by a positive number:** You can multiply or divide both sides of an inequality by a positive number without changing the inequality sign.
  • **Multiplying or dividing by a negative number:** When you multiply or divide both sides by a negative number, you must switch the direction of the inequality sign (\(<\) becomes \(>\) and vice versa). This is crucial as not doing so would result in an incorrect solution.
  • **Combining inequalities:** When dealing with compound inequalities (e.g., \(a < x < b\)), ensure operations done on all parts are consistent to maintain truth.
By understanding and applying these fundamental rules, you'll be more adept at solving inequalities correctly and efficiently.

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 room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 654 , find the classroom numbers.

A quadrilateral is a polygon with 4 sides. The sum of the measures of the 4 angles in a quadrilateral is \(360^{\circ} .\) If the measures of the angles of a quadrilateral are consecutive odd integers, find the measures.

On an NBA team the two forwards measure \(6^{\prime} 8^{\prime \prime}\) and \(6^{\prime} 6^{\prime \prime}\) tall and the two guards measure \(6^{\prime} 0^{\prime \prime}\) and \(5^{\prime} 9^{\prime \prime}\) tall. How tall should the center be if they wish to have a starting team average height of at least \(6^{\prime} 5^{\prime \prime} ?\)

One more than five times a number is less than or equal to ten. Find all such numbers.

The area of the Sahara Desert is 7 times the area of the Gobi Desert. If the sum of their areas is 4,000,000 square miles, find the area of each desert.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.