Solving Linear Inequalities in One Variable - 2 | 8. Linear Inequalities | IB Class 10 Mathematics – Group 5, Algebra
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Academics

Grades

Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Curriculum

CBSE ICSE IB
Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Professional Courses
Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.

games
Typing
Memory
Math
English Adventures
Knowledge

2 - Solving Linear Inequalities in One Variable

Enroll to start learning

You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take mock test.

Practice

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to Linear Inequalities

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Today, we’re discussing linear inequalities in one variable. Can anyone tell me how they differ from regular equations?

Student 1
Student 1

Linear inequalities use symbols like < and > instead of just =.

Teacher
Teacher

Exactly! While equations show equality between two expressions, inequalities compare them, resulting in a range of values. Now, let's look at how to solve these inequalities. What do we need to remember?

Student 2
Student 2

We have to reverse the sign when we multiply or divide by a negative number!

Teacher
Teacher

Great memory! Remember, we use the acronym 'RIF'—Reverse If Negative—to help us remember. Let's try a sample inequality together.

Student 3
Student 3

Like `2x - 3 < 5`?

Teacher
Teacher

Exactly! How do we tackle that one?

Student 4
Student 4

First, we add 3 to both sides, giving us `2x < 8`.

Teacher
Teacher

Perfect! And what’s the next step?

Student 1
Student 1

We divide by 2, so `x < 4`!

Teacher
Teacher

Well done! Let’s summarize: Linear inequalities use different symbols and require special rules when we multiply or divide by negatives. Let’s move on to graphing these solutions.

Graphing Linear Inequalities

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Now, let’s talk about graphing our solutions. Can someone remind me how we represent the solutions on a number line?

Student 2
Student 2

We use open circles for < or > and closed circles for ≤ or ≥.

Teacher
Teacher

Excellent! And what does shading indicate?

Student 3
Student 3

It shows the range of solutions that satisfy the inequality!

Teacher
Teacher

Correct! Let’s visualize this using the example we solved earlier, `x < 4`. How would that look?

Student 4
Student 4

We place an open circle on 4 and shade everything to the left!

Teacher
Teacher

Exactly! This gives us a complete visual representation of the solutions. Always remember to check the direction of the inequality as you shade.

Student 1
Student 1

So, the open circle means 4 isn’t included?

Teacher
Teacher

Yes! Now let’s move to the next session and explore examples with inequalities containing negative coefficients.

Solving with Negative Coefficients

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

0:00
Teacher
Teacher

Now, let’s discuss solving inequalities that involve negatives. For instance, how do we solve `-3x + 4 ≥ 10`?

Student 2
Student 2

We start by subtracting 4 from both sides to get `-3x ≥ 6`.

Teacher
Teacher

Correct! And what happens next?

Student 3
Student 3

We divide by -3, so we must reverse the inequality, leading to `x ≤ -2`.

Teacher
Teacher

Exactly! This important step is crucial whenever you deal with negatives. Can someone summarize our key points from today?

Student 4
Student 4

Always remember to reverse the sign when dividing by a negative and graph with open or closed circles depending on the symbol!

Teacher
Teacher

Great summary! Now you have an excellent foundation for solving and graphing linear inequalities. Let's wrap up with some practical exercises.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explains how to solve linear inequalities in one variable, highlighting key concepts such as reversing inequality signs during multiplication or division by negative numbers and graphing solutions on a number line.

Standard

In this section, we will explore linear inequalities in one variable, covering essential rules for solving them, such as the need to reverse the inequality sign when multiplying or dividing by a negative number. We will also discuss how to represent solutions graphically on a number line using open and closed circles.

Detailed

Detailed Summary

This section focuses on solving linear inequalities in one variable, which operates under similar principles as linear equations but requires additional consideration in certain operations. Here are the key points discussed:

  1. Definition of Linear Inequalities: Linear inequalities utilize symbols such as <, ≤, >, and ≥ to indicate the relationship between two expressions, yielding a range of possible solutions rather than a single fixed answer.
  2. Rules for Solving Inequalities: Solving linear inequalities involves performing operations (addition, subtraction, multiplication, or division) just like solving equations. However, a critical rule is that when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
  3. Examples: This section includes examples with step-by-step solutions to clarify the process. For instance:
  4. For the inequality 2x - 3 < 5, the solution leads to x < 4.
  5. For -3x + 4 ≥ 10, after correctly flipping the inequality sign, the result is x ≤ -2.
  6. Graphing Solutions: When graphing on a number line, an open circle represents inequalities that do not include the endpoint (for < or >), while a closed circle represents those that include it (for ≤ or ≥). Additionally, shading indicates the portion of the number line that satisfies the inequality.

This foundational understanding of linear inequalities sets the stage for grappling with more complex concepts later in the chapter, including inequalities in two variables.

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Rules to Remember

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

• You can add, subtract, multiply, or divide both sides of an inequality by the same number except:
o When multiplying or dividing by a negative number, reverse the inequality sign.

Detailed Explanation

In solving linear inequalities, it's crucial to remember that basic algebraic operations can be performed on both sides of the inequality just like in equations. However, there is one significant exception: if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This means, for example, if you have -x > 3 and you divide both sides by -1, it becomes x < -3.

Examples & Analogies

Think of this like a game where you're trying to maintain the same relationship while changing teams. If you have an agreement (inequality) and you change the 'team' (multiply/divide) to the opposing side (negative), the terms of the agreement must also change. Just like if you're negotiating an advantage in a game, flipping the terms helps you stay fair!

Examples of Solving Inequalities

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

🔹 Examples:
1. Solve: 2x - 3 < 5
→ 2x < 8
→ x < 4
2. Solve: -3x + 4 ≥ 10
→ -3x ≥ 6
→ x ≤ -2 (Notice how the sign flips)

Detailed Explanation

Let's break down the examples given:
1. In the first example, we start with 2x - 3 < 5. To isolate x, we first add 3 to both sides, yielding 2x < 8. Dividing both sides by 2 results in x < 4. Thus, any value less than 4 makes the inequality true.
2. In the second example, we see -3x + 4 ≥ 10. First, we subtract 4 from both sides to get -3x ≥ 6. Now, we divide by -3; since we're dividing by a negative, the inequality sign flips, giving us x ≤ -2. This indicates that x can be any number less than or equal to -2.

Examples & Analogies

Imagine you're at a race with a friend. If you tell them they cannot go more than 4 meters ahead of you, that's like saying x < 4. If your friend tries to cross that line (negative side), you have to adjust your agreement (flip the sign) to ensure it’s fair.

Graphing on the Number Line

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

• Use an open circle (○) for < or >.
• Use a closed circle (●) for ≤ or ≥.
• Shade to the left or right based on the inequality direction.

Detailed Explanation

When graphing linear inequalities on a number line, you represent the solution set visually. An open circle at a number (e.g., x < 4) shows that the number itself is not included in the solution, while a closed circle (e.g., x ≤ -2) indicates that the number is included. After marking the circle, you then shade the line to the left or right depending on whether the solution represents numbers less than (shade left) or greater than (shade right) that point.

Examples & Analogies

Visualize a road where some paths are open and others are closed based on your restrictions. If you're telling someone they can only go left (x < 4), you mark an open path (open circle) at the divider, while if you've told them they can come to your house (x ≤ -2), you would mark it as included (closed circle) on the road.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Linear Inequalities: Use symbols to represent ranges of possible solutions.

  • Sign Reversal: Necessary when multiplying or dividing by negative numbers.

  • Graphing Techniques: Open and closed circles represent inclusion and exclusion.

  • Shading Solutions: Indicates the range of values that satisfy the inequality.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: Solve 2x - 3 < 5: Add 3 to both sides to get 2x < 8, then divide by 2 to find x < 4.

  • Example 2: Solve -3x + 4 ≥ 10: Subtract 4 from both sides to get -3x ≥ 6, then divide by -3, reversing the sign, results in x ≤ -2.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • When a negative's in the game, flip the sign to keep the same!

📖 Fascinating Stories

  • Imagine a seesaw: when you push down (negative), the other side must lift up (reverse the sign) to balance it out.

🧠 Other Memory Gems

  • RIF - Remember If Negative: Flip the sign!

🎯 Super Acronyms

GLOW - Graphing Line with Open or closed circle, depending on the sign.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Linear Inequality

    Definition:

    An inequality that involves a linear expression and uses symbols such as <, ≤, >, or ≥.

  • Term: Solution Set

    Definition:

    The range of values that satisfy a given inequality.

  • Term: Open Circle

    Definition:

    A symbol used on a number line to indicate that a point is not included in the solution set (used for < or >).

  • Term: Closed Circle

    Definition:

    A symbol used on a number line to indicate that a point is included in the solution set (used for ≤ or ≥).

  • Term: Inequality Sign Flip

    Definition:

    The necessity to reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number.