The Complete Guide to Inequalities

Inequality is a mathematical expression that expresses the relationship between the magnitude of two quantities and is one of the compulsory topics in DSE Mathematics. Mastering the solution of inequalities not only improves your maths performance, but also develops logical thinking skills. In this article, we will explain the techniques of solving primary, compound and quadratic inequalities, and combine them with real-life examples from the DSE exam to help you tackle all types of inequality problems easily.

Graphical representation of an inequality and its solution.

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What is an inequality? --Basic Concepts and Symbols

Inequality is a mathematical term for the use of the symbols ">", "¡", "¡", "¡", "¡", "¡" and "¡".

Basically unequal numbers:

  • Greater than (>):Indicates that the value on the left is greater than the value on the right.
  • Less Than (indicates that the value on the left is less than the value on the right)
  • Greater than or equal to (≥):Indicates that the value on the left is greater than or equal to the value on the right.
  • Less than or equal to (≤):Indicates that the value on the left is less than or equal to the value on the right.
  • Not equal to (≠):Indicates that the values on both sides are not equal

Examples of inequalities in life:

  • Age limit: "must be aged 18 or above" can be written as Age ≥ 18.
  • Requirement: "Maths score over 80" can be written as Maths score > 80.
  • Weight limit: "Baggage weight not exceeding 20kg" can be written as Weight ≤ 20
Basic Inequality Symbols and Linear Representation

Differences from equations:An equation means that the two sides are exactly equal, while an inequality means that the two sides are related by size. The solution to an inequality is usually a range rather than a single value.

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Primary Inequality - Solution Procedure and Line Representation

A quadratic inequality is the most basic type of inequality and takes the form ax + b > 0 (or

Basic Steps for Solving Problems

  1. Move items: move items containing unknowns to one side, and constant items to the other side
  2. Combining Similar Terms: Simplifying Inequalities
  3. Coefficient to 1: Divide both sides by the coefficient of the unknown.
  4. Note the direction of the sign: when dividing by a negative number, the inequality sign changes direction.

Important: When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign must change! This is one of the most common mistakes students make.

Importance

  • If both sides of an inequality are added or subtracted at the same time, the direction of the inequality sign remains unchanged.
  • When both sides of an inequality are multiplied or divided by the same positive number, the direction of the inequality sign remains unchanged.
  • When both sides of an inequality are multiplied or divided by the same negative number, the direction of the inequality sign changes.
Flowchart for solving one-variable inequalities

Example: Solving the inequality 3x - 5 > 7

Solution:

3x - 5 > 7

3x > 7 + 5 (shift)

3x > 12 (combined like-for-like)

x > 4 (both sides divided by 3)

Answer:x > 4

Example: Solving the inequality -2x + 3 ≤ 9

Solution:

-2x + 3 ≤ 9

-2x ≤ 9 - 3 (shift)

-2x ≤ 6 (combine similar items)

x ≥ -3 (both sides divided by -2, inequality sign changes direction)

Answer:x ≥ -3

Linear representation

The number line representation is an important method for visualising the solution of inequalities:

  • Open area (> or indicated by a hollow circle to exclude the point)
  • Closed range (≥ or ≤):A solid dot indicates that the point is included.
  • Arrow direction:Points to a range of values that satisfy the inequality
Representation of inequalities on a number line

Complex Inequalities - "And" and "Or" Logic and Problem Solving Strategies

Compound inequalities are made up of two or more inequalities connected by the logical conjunctions "and" or "or". Understanding these logical relationships is essential to solving problems.

The case of "and" (intersection)

The two inequalities must hold together, and the solution is the intersection (overlapping part) of the two solution sets.

For example, x > 2 and x

The case of "or" (clustering)

At least one of the two inequalities holds and the solution is the union of the two solution sets (all ranges).

For example, if x 3, the solution is x 3

Tips for solving problems:In solving compound inequalities, each inequality is solved separately, then the set of solutions is marked on the number line, and the final solution is found by using the logical relationship of "and" or "or".

Complex Inequality Graphs and Lines

Example: The case of "and

Title:Solution 2x - 5 > 11 and -3x

Solution:

Solve the first inequality first: 2x - 5 > 11 → 2x > 16 → x > 8

Solve the second inequality: -3x -3 (note the change in direction of the inequality sign)

"And" means that two conditions hold: x > 8 and x > -3.

Answer:x > 8

Example: The case of "or

Title:Solution x + 3 7

Solution:

Solve the first inequality first: x + 3

Solve the second inequality: 2x - 1 > 7 → 2x > 8 → x > 4

"Or" means at least one condition holds: x 4

Answer:x 4

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Quadratic Inequalities - Algebraic and Graphical Methods

A quadratic inequality is of the form ax² + bx + c > 0 (or

Algebraic method (factorisation)

  1. Factorise a quadratic formula
  2. Find the root that makes each factor zero
  3. Analysing the positivity and negativity of districts using a table of symbols or a number line.
  4. Select the appropriate range according to the inequality sign.

Example: Algebraic Solution

Title:Solving the inequality x² - x - 12 > 0

Solution:

Step 1: Factorise x² - x - 12 = (x - 4)(x + 3)

Step 2: Find the root x = 4 or x = -3.

Step 3: Analyse the symbols between districts:

  • x 0
  • -3 0
  • x > 4: (+)(+) = (+), satisfaction > 0

Answer:x 4

Diagramming

  1. Graph a quadratic function (parabola)
  2. Find the intersection with the x-axis (root)
  3. Select the area above or below the x-axis according to the inequality sign.
Graphical solution of quadratic inequalities in one variable

Example: Graphical Solution

Title:Solve the inequality -x² + 2x + 3 ≥ 0

Solution:

Step 1: Factorise -x² + 2x + 3 = -(x² - 2x - 3) = -(x - 3)(x + 1)

Step 2: Find the root x = 3 or x = -1.

Step 3: Since a = -1

Step 4: ≥ 0 means select the part above the x-axis (including the x-axis).

Answer:-1 ≤ x ≤ 3

Judgement of "inward" and "outward" orientation

Parabola with opening upwards (a > 0)

  • f(x) > 0:Select the part of the parabola that is "above" the x-axis (outward)
  • f(x) Select the part of the parabola that is "below" the x-axis (inward)

Open downward parabola (a)
  • f(x) > 0:Select the part of the parabola that is "above" the x-axis (inwards).
  • f(x) Select the part of the parabola that is "below" the x-axis (outward)

Inward and Outward Judgements of Quadratic Inequalities

Properties of Inequalities - Rules and Precautions

Additive Nature

If a > b, then a + c > b + c.

When both sides of an inequality are added or subtracted at the same time, the direction of the inequality sign remains unchanged.

Multiplicative properties (positive)

If a > b and c > 0, then ac > bc.

When both sides of an inequality are multiplied or divided by the same positive number, the direction of the inequality sign remains unchanged.

Multiplicative nature (negative numbers)

If a > b and c

When both sides of an inequality are multiplied or divided by the same negative number, the direction of the inequality sign changes.

transferability

If a > b and b > c, then a > c.

Inequality relations are transitive and allow for chain reasoning.

Inverse nature

If a > b > 0, then 1/a

For positive numbers, taking the inverse will change the direction of the inequality sign.

square metric tonnes

If a > b ≥ 0, then a² > b².

For non-negative numbers, the squaring operation maintains the direction of the inequality sign.

Common Mistakes and Points to Note

❌ Forget to change the direction of the inequality sign

When both sides are multiplied or divided by a negative number, the direction of the inequality sign must be changed.

Error:-2x > 6 → x > -3

Correct:-2x > 6 → x

❌ Confusing the closed area

Note the difference between > and ≥, which affects the representation on the number line.

x > 2:Use hollow circles, not containing 2

x ≥ 2:Use solid dots containing 2

❌ Compound Inequality Logic Error

Correctly understand the logical relationship between "and" and "or".

"and":Taking the intersection (overlapping parts)

"or":Fetch set (all ranges)

Comparison of common mistakes and correct solutions of inequalities.

Inequalities in DSE Maths - Exam Tips and Examples

Inequalities are an important topic in DSE Maths (Compulsory Part) and appear frequently in the exam. The following tips and examples will help you do well in the exam.

Types of Inequality Questions in DSE Exam

  1. Basic solutions to one-variable inequalities
  2. Logical Reasoning for Compound Inequalities
  3. Graphical and Algebraic Solutions of Quadratic Inequalities
  4. Application of Inequalities in Practical Problems
  5. Combined questions with function images

Exam Solving Skills

  • Read the questions carefully:Confirm the direction of the inequality sign and whether it contains an equal sign.
  • Choose the appropriate method:Choose algebraic or graphical methods according to the characteristics of the problem.
  • Check the answer:Substitute the solution into the original inequality to verify that
  • Be aware of special circumstances:Such as unresolved, constant, etc.

GETUTOR Expert Tip:In the DSE exam, inequality questions are often combined with real-world application scenarios, such as maximising profits and minimising costs. Understanding the relationship between inequalities and functions will help you to better solve such questions.

Inequality questions in DSE Maths Exam

DSE Style Example

Example question:The relationship between a shop's monthly sales y ($10,000) and advertising expenses x ($10,000) can be expressed as a function y = -x² + 8x + 20, where 0 ≤ x ≤ 10.

(a) What is the cost of advertising to maximise monthly sales?

(b) What is the range of advertising costs if monthly sales are to exceed $350,000?

Answer:

(a) y = -x² + 8x + 20 = -(x² - 8x) + 20 = -(x - 4)² + 16 + 20 = -(x - 4)² + 36

When x = 4, y has a maximum value of $360,000.

(b) To make y > 35, i.e. -x² + 8x + 20 > 35

-x² + 8x + 20 - 35 > 0

-x² + 8x - 15 > 0

x² - 8x + 15

(x - 3)(x - 5)

Therefore 3

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Time Management Recommendations

Type of inequality Suggested Solution Time Key Points
one-dimensional inequality 1-2 minutes Note the handling of shifts and coefficients, especially in the case of negative numbers.
Compound inequality 3-4 minutes To distinguish the logical relationship between "and" and "or", and to take the intersection or union correctly.
quadratic inequality 4-6 minutes To master factorisation, paying attention to the judgement of "inward" and "outward".
Application Topics 6-8 minutes Develop correct mathematical models and explain the answers in real-life situations

Practice Questions and Explanations - Consolidate what you've learnt

Test your understanding of inequalities with the following practice questions:

Exercise 1

Title:Solving the inequality 3(x - 2) - 2(x + 1) > x - 8

View Answers

3(x - 2) - 2(x + 1) > x - 8

3x - 6 - 2x - 2 > x - 8

x - 8 > x - 8

0 > 0

This is a contradiction, so the original inequality is not solved.

Exercise 2

Title:Solve the complex inequality -1 ≤ 2x + 3

View Answers

-1 ≤ 2x + 3

Decompose into two inequalities:

-1 ≤ 2x + 3 and 2x + 3

Solve the first one: -1 ≤ 2x + 3 → -4 ≤ 2x → x ≥ -2

Solve the second one: 2x + 3

Answer: -2 ≤ x

Exercise 3

Title:Solve the quadratic inequality x² - 5x + 6 ≤ 0

View Answers

x² - 5x + 6 ≤ 0

Factorisation: (x - 2)(x - 3) ≤ 0

The roots are x = 2 and x = 3

Analyse the districts:

x 0, unsatisfactory

2 ≤ x ≤ 3: (-)(+) = (-)

x > 3: (+)(+) = (+) > 0, unsatisfactory

Answer: 2 ≤ x ≤ 3

Inequality Practice Problems Analysis and Line Representation

Mastering Inequality Solving to Improve Mathematics Achievement!

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Calculator Tips - Solving Inequality Problems Quickly

Proficient use of calculators can improve the efficiency of problem solving in the DSE exam, especially when dealing with complex quadratic inequalities.

CASIO fx-50FH II / fx-991EX

  1. Press [MODE] → [5] (EQN) → [2] (ax²+bx+c=0)
  2. Enter the coefficients a, b, c in that order.
  3. Press [=] to get two roots
  4. Using roots to analyse solutions to inequalities
CASIO Calculator Steps to Solve Inequalities

TI-84 Plus

  1. Press [Y=] to enter the function.
  2. Press [GRAPH] to view the image
  3. Use [2nd] [CALC] for change.
  4. Determine the solution to the inequality based on the image.
TI-84 Calculator Steps to Solve Inequalities

Tips for using the calculator

  • Save time on exams by mastering basic operations
  • Use graphical functions to understand solutions to inequalities intuitively
  • Use the table function to verify the correctness of the solution.
  • Be aware of the limitations of the calculator's accuracy and perform manual validation if necessary

Summary - Key Points for Solving Inequalities

Inequality is an important concept in Mathematics and is a compulsory part of the DSE exam. You should have learnt it through this article:

  • Basic Concepts and Symbols of Inequalities
  • Steps for solving one-variable inequalities
  • The "and" and "or" logic of compound inequalities
  • Algebraic and graphical methods of quadratic inequalities
  • Importance Properties of Inequalities and Rules of Arithmetic
  • Inequality Question Types and Solving Techniques in DSE Exam

Continuous practice is the key to mastering the solution of inequalities. It is recommended that you do more practice problems and try to apply what you have learnt to real-world problems.

Summary and Application of Inequality Solutions

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