01-03 - Linear Inequalities

Phase: 1 | Subject: 01-03 Prerequisites: 01-02-linear-equations.md (solving linear equations) Next subject: 01-04-coordinate-geometry-2d.md

Learning Objectives

By the end of this subject, you will be able to:

1. Interpret and write inequalities using correct notation (<, >, ≤, ≥)
2. Represent inequalities on a number line
3. Solve linear inequalities using the same techniques as equations
4. Handle the sign-flip rule when multiplying or dividing by a negative
5. Solve compound inequalities and express answers in interval notation

Core Content

Inequality Notation

Key difference from equations: Inequalities have a RANGE of solutions, not a single value.

Number Line Representation

• Open circle (< or >): the endpoint is NOT included
• Closed circle (≤ or ≥): the endpoint IS included
• Arrow points in the direction of the solution set

$x < 3      x > -2      x ≤ 5      x ≥ -1
---o======  ====o----  ----[====  ====]----
  0          -2  0  5    0
$

Solving Linear Inequalities

The process is IDENTICAL to solving equations, with ONE critical exception.

Example: $2x + 5 > 11$

1. Subtract 5: 2x > 6
2. Divide by 2: x > 3

Check: Try x = 4: $2(4) + 5 = 13 > 11$ ✓ Try x = 2: $2(2) + 5 = 9 > 11$ ✗ (9 is NOT greater than 11)

The Sign-Flip Rule

CRITICAL: When you multiply or divide BOTH SIDES by a NEGATIVE number, you MUST flip the inequality sign.

Why? Think about 3 > 1. If we multiply both sides by -1: - Incorrect: $-3 > -1$ (FALSE - on the number line, -3 is to the LEFT of -1) - Correct: $-3 < -1$ (TRUE - the order reverses when we reflect through 0)

Example: $-2x > 8$

1. Divide by -2 (NEGATIVE!)
2. FLIP the sign: $x < -4$

Check: Try x = -5: $-2(-5) = 10 > 8$ ✓ Try x = -3: $-2(-3) = 6 > 8$ ✗

Compound Inequalities

"And" inequalities (intersection)

$-2 < x ≤ 5$ means x must satisfy BOTH conditions. On a number line: open circle at -2, closed circle at 5, shaded between.

Example: Solve $1 < 2x + 3 ≤ 9$

1. Subtract 3 everywhere: $-2 < 2x ≤ 6$
2. Divide by 2: $-1 < x ≤ 3$

"Or" inequalities (union)

$x < -2 OR x > 5$ means x satisfies EITHER condition. On a number line: shaded left of -2 AND shaded right of 5.

Interval Notation

A compact way to write solution sets:

Parentheses () = not included Square brackets [] = included Always use ∞ with parentheses (you can't "reach" infinity)

Key Terms

• "And" inequalities (intersection)
• "Or" inequalities (union)
• 01 03 Linear Inequalities
• Always use ∞ with parentheses
• Compound Inequalities
• Correct: A)
• Correct: B)
• Example 1: Solve 3x - 7 ≥ 2x + 5
• Example 2: Solve -4x + 3 < 11
• Example 3: Solve -3 ≤ 2x - 5 < 7
• Inequality
• Inequality Notation

Worked Examples

Example 1: Solve 3x - 7 ≥ 2x + 5

1. Move x terms to left: $3x - 2x - 7 ≥ 5$
2. Simplify: $x - 7 ≥ 5$
3. Add 7: $x ≥ 12$

In interval notation: $[12, ∞)$

Check: x = 12: $3(12) - 7 = 36 - 7 = 29$ and $2(12) + 5 = 24 + 5 = 29$ ✓ x = 13: $3(13) - 7 = 39 - 7 = 32$ and $2(13) + 5 = 26 + 5 = 31$. $32 ≥ 31$ ✓

Example 2: Solve -4x + 3 < 11

1. Subtract 3: $-4x < 8$
2. Divide by -4 (NEGATIVE - FLIP!): $x > -2$

In interval notation: $(-2, ∞)$

Check: x = 0: $-4(0) + 3 = 3 < 11$ ✓ x = -3: $-4(-3) + 3 = 12 + 3 = 15 < 11$ ✗ (correctly excluded)

Example 3: Solve -3 ≤ 2x - 5 < 7

1. Add 5 everywhere: $2 ≤ 2x < 12$
2. Divide by 2: $1 ≤ x < 6$

In interval notation: [1, 6)

Check: x = 1: $-3 ≤ 2(1) - 5 = -3 < 7$ ✓ x = 6: $-3 ≤ 2(6) - 5 = 7 < 7$ ✗ (7 is not less than 7, correctly excluded)

Quiz

Q1: What does the concept of Compound Inequalities primarily refer to in this subject?

A) A visual representation of Compound Inequalities B) A computational error related to Compound Inequalities C) A historical anecdote about Compound Inequalities D) The definition and application of Compound Inequalities

Correct: D)

• If you chose A: This is incorrect. Compound Inequalities is defined as: the definition and application of compound inequalities. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Compound Inequalities is defined as: the definition and application of compound inequalities. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Compound Inequalities is defined as: the definition and application of compound inequalities. The other options describe different aspects that are not the primary focus.
• If you chose D: Compound Inequalities is defined as: the definition and application of compound inequalities. The other options describe different aspects that are not the primary focus. Correct!

Q2: What is the primary purpose of Inequality?

A) It is primarily a historical notation system B) It is used to inequality in mathematical analysis C) It is used only in advanced research contexts D) It replaces all other methods in this domain

Correct: B)

• If you chose A: This is incorrect. Inequality serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Inequality serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Inequality serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Inequality serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Inequality Notation is TRUE?

A) Inequality Notation is an advanced topic beyond this subject's scope B) Inequality Notation is mentioned only as a historical footnote C) Inequality Notation is a fundamental concept covered in this subject D) Inequality Notation is not related to this subject

Correct: C)

• If you chose A: This is incorrect. Inequality Notation is a fundamental concept covered in this subject. This subject covers Inequality Notation as part of its core content.
• If you chose B: This is incorrect. Inequality Notation is a fundamental concept covered in this subject. This subject covers Inequality Notation as part of its core content.
• If you chose C: Inequality Notation is a fundamental concept covered in this subject. This subject covers Inequality Notation as part of its core content. Correct!
• If you chose D: This is incorrect. Inequality Notation is a fundamental concept covered in this subject. This subject covers Inequality Notation as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) An unrelated numerical value C) A different result from a common mistake D) ** Inequalities have a RANGE of solutions, not a s

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is ** Inequalities have a RANGE of solutions, not a s. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is ** Inequalities have a RANGE of solutions, not a s. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is ** Inequalities have a RANGE of solutions, not a s. The other options represent common errors.
• If you chose D: The worked examples show that the result is ** Inequalities have a RANGE of solutions, not a s. The other options represent common errors. Correct!

Q5: How are Inequality Notation and Number Line Representation related?

A) Inequality Notation and Number Line Representation are completely unrelated topics B) Inequality Notation is the inverse of Number Line Representation C) Inequality Notation is a special case of Number Line Representation D) Inequality Notation and Number Line Representation are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Inequality Notation and Number Line Representation are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Inequality Notation and Number Line Representation are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Inequality Notation and Number Line Representation are covered in this subject as interconnected topics.
• If you chose D: Both Inequality Notation and Number Line Representation are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Solving Linear Inequalities?

A) A common mistake is confusing Solving Linear Inequalities with a similar concept B) Solving Linear Inequalities is always computed the same way in all contexts C) The main error with Solving Linear Inequalities is using it when it is not needed D) Solving Linear Inequalities has no common misconceptions

Correct: A)

• If you chose A: Students often confuse Solving Linear Inequalities with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Solving Linear Inequalities with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Solving Linear Inequalities with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Solving Linear Inequalities with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply The Sign-Flip Rule?

A) The Sign-Flip Rule is not practically useful B) Apply The Sign-Flip Rule to solve problems in this subject's domain C) Avoid The Sign-Flip Rule unless explicitly instructed D) Use The Sign-Flip Rule only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. The Sign-Flip Rule is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: The Sign-Flip Rule is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. The Sign-Flip Rule is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. The Sign-Flip Rule is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Solve: $x - 4 > 9$
   Click for answer

x > 13, or $(13, ∞)$

1. Solve: $5 - 2x ≤ 11$
   Click for answer

$-2x ≤ 6$, divide by -2 (flip!): $x ≥ -3$, or $[-3, ∞)$

1. Solve: $3x + 2 ≥ 5x - 4$
   Click for answer

$4 ≥ 2x$, so $2 ≥ x$, or $x ≤ 2$, or $(-∞, 2]$

1. Solve: $-x + 3 < 7$
   Click for answer

$-x < 4$, multiply by -1 (flip!): $x > -4$, or $(-4, ∞)$

1. Solve: $-2 ≤ 3x + 1 < 10$
   Click for answer

$-3 ≤ 3x < 9$, so $-1 ≤ x < 3$, or $[-1, 3)$

1. Solve: $(x + 1)/4 ≥ (x - 2)/3$
   Click for answer

Cross multiply (3 > 0, no flip): $3(x + 1) ≥ 4(x - 2)$, $3x + 3 ≥ 4x - 8$, $11 ≥ x$, or $x ≤ 11$

1. Which values satisfy $-3 < 2x - 1 ≤ 5$?
   Click for answer

$-2 < 2x ≤ 6$, so $-1 < x ≤ 3$, or $(-1, 3]$

Summary

Key takeaways:

• Inequalities have ranges of solutions, not single values
• Use the same algebraic techniques as equations
• ALWAYS flip the sign when multiplying/dividing by a negative number
• Compound "and" inequalities require both conditions (intersection)
• Compound "or" inequalities require either condition (union)
• Interval notation: parentheses = not included, brackets = included
• Always verify your solution by testing boundary points

Pitfalls

• Forgetting to flip the sign when multiplying or dividing by a negative. This is the number one mistake with inequalities. If you divide both sides of -2x > 8 by -2, you must write x < -4, not x > -4. Always pause and ask: "Did I just multiply or divide by a negative?"
• Dropping the inequality direction when rearranging. When solving 3x + 2 ≥ 5x - 4, if you move terms incorrectly, you can lose or flip the inequality unintentionally. Keep the inequality symbol the same unless you multiply/divide by a negative.
• Using the wrong bracket in interval notation. x ≥ -2 is [-2, ∞) with a square bracket, not (-2, ∞) with a parenthesis. The square bracket means the endpoint is INCLUDED. Parentheses mean excluded. This distinction is critical and frequently mixed up.
• Misreading compound inequalities. The expression -3 ≤ 2x - 5 < 7 means both conditions must hold simultaneously. Students often solve only one side or forget that the middle expression is shared. Perform the same operation on ALL THREE parts.
• Forgetting to check boundary points. Testing one value inside the solution set isn't enough — test the boundary (where x equals the endpoint) to confirm whether it's included or excluded, especially for ≤ and ≥.

Next Steps

Next up: 01-04-coordinate-geometry-2d.md