Inequalities and absolute values

Overview

Inequalities compare numbers that are not necessarily equal. Absolute value measures how far a number is from zero on the number line. Together, these ideas help learners compare quantities, describe ranges, and reason about positive and negative numbers.

For example, $-2<3$ because $-2$ lies to the left of $3$ on the number line. Also, $|-2|=2$ because $-2$ is $2$ units from zero.

The two ideas support each other. Inequalities tell which side a number is on or which number is larger. Absolute value tells how far a number is from $0$, without caring whether the number is left or right of $0$.

Prerequisites

• Ordering numbers on a number line.
• Positive and negative integers.
• Fractions and decimals as rational numbers.
• Understanding distance as a non-negative measure.
• Basic use of $=$ for equality.

Learning Scope

This page covers inequality symbols, comparing real numbers, representing simple inequalities on a number line, and evaluating absolute values.

It does not fully cover solving algebraic inequalities in one unknown, simultaneous inequalities, or linear programming. Those appear in later algebra pages such as Inequalities in one unknown and Linear programming.

Subtopics

Inequality Symbols

The main inequality symbols are:

• $<$ means "less than".
• $>$ means "greater than".
• $\le$ means "less than or equal to".
• $\ge$ means "greater than or equal to".
• $\ne$ means "not equal to".

Key insight: the open side of $<$ or $>$ faces the greater number.

Examples:

$$ 2<5,\quad 7>1,\quad -3<0 $$

The symbols $\le$ and $\ge$ include equality. For example, $x\le 4$ allows $x$ to be $4$, but $x<4$ does not. This small difference is important in number-line diagrams and word problems.

Comparing Numbers on a Number Line

Numbers increase from left to right. If $a$ is to the left of $b$, then $a<b$. If $a$ is to the right of $b$, then $a>b$.

For example:

$$ -6<-2 $$

because $-6$ is farther left than $-2$.

Key insight: for negative numbers, the number with the larger distance from zero is smaller.

A useful slow method is to ask, "Which point is farther right?" The farther-right point is the greater number. This works for positive numbers, negative numbers, fractions, and decimals.

Inequalities With Fractions and Decimals

Fractions and decimals can be compared by converting to a common form.

For example:

$$ \frac{3}{4}=0.75 $$

Since $0.75>0.7$, then:

$$ \frac{3}{4}>0.7 $$

Another method is to convert both values to fractions with a common denominator:

$$ 0.7=\frac{7}{10},\quad \frac{3}{4}=\frac{15}{20},\quad \frac{7}{10}=\frac{14}{20} $$

So $\frac{3}{4}$ is greater. The method chosen should make the comparison easier, not longer.

Simple Inequality Statements

An inequality may describe a set of numbers. For example:

$$ x>2 $$

means all numbers greater than $2$. On a number line, this is shown to the right of $2$. If the endpoint is not included, use an open circle. If the endpoint is included, as in $x\ge 2$, use a closed circle.

Direction matters:

• $x>2$ is shaded to the right of $2$.
• $x<2$ is shaded to the left of $2$.
• $x\ge 2$ includes $2$ and goes right.
• $x\le 2$ includes $2$ and goes left.

At Form I level, learners should first read and represent simple inequalities before moving to solving algebraic inequalities.

Absolute Value

The absolute value of a number is its distance from $0$ on the number line.

$$ |a|=\text{distance of }a\text{ from }0 $$

Examples:

$$ |6|=6,\quad |-6|=6,\quad |0|=0 $$

Key insight: absolute value is never negative.

Absolute value answers a distance question: "How many units from zero?" For this reason, $|-10|$ is $10$, not $-10$. The negative sign tells the original position, but the distance is positive.

Absolute Value and Opposites

Opposite numbers have the same absolute value.

$$ |8|=|-8|=8 $$

This does not mean $8=-8$. It means both numbers are the same distance from $0$.

In general:

$$ |a|=|-a| $$

for any number $a$. The two numbers are opposites unless $a=0$, but their distances from $0$ are equal.

Key Terms

• Inequality: a statement comparing quantities using symbols such as $<$, $>$, $\le$, or $\ge$.
• Less than: located to the left on a number line.
• Greater than: located to the right on a number line.
• Endpoint: the boundary value in an inequality.
• Absolute value: distance of a number from zero.
• Opposite numbers: numbers with the same distance from zero but different signs.
• Open circle: a number-line mark showing an endpoint is not included.
• Closed circle: a number-line mark showing an endpoint is included.

Worked Examples

Example 1: Complete inequality symbols

Insert $<$ or $>$ between $-4$ and $1$.

Since $-4$ is to the left of $1$ on the number line:

$$ -4<1 $$

Check using words: "$-4$ is less than $1$." The sentence is true, so the symbol is correct.

Example 2: Compare fractions and decimals

Compare $\frac{5}{8}$ and $0.6$.

Convert the fraction to a decimal:

$$ \frac{5}{8}=0.625 $$

Now compare:

$$ 0.625>0.6 $$

Therefore:

$$ \frac{5}{8}>0.6 $$

If using fractions instead:

$$ 0.6=\frac{6}{10}=\frac{3}{5} $$

Then compare $\frac{5}{8}$ and $\frac{3}{5}$:

$$ 5 \times 5=25,\quad 8 \times 3=24 $$

So $\frac{5}{8}>\frac{3}{5}$.

Example 3: Evaluate absolute values

Find $|-9|+|4|$.

$$ \begin{aligned} |-9|+|4| &= 9+4 \\ &= 13 \end{aligned} $$

The answer is $13$.

Notice that the absolute value signs are handled before adding. Do not add $-9+4$ first, because $|-9|+|4|$ is not the same as $|-9+4|$.

Example 4: Interpret a simple inequality

Describe the numbers represented by $x\le -2$.

The symbol $\le$ means "less than or equal to". Therefore, the solution includes $-2$ and all numbers to the left of $-2$ on the number line.

Examples include:

$$ -2,\quad -3,\quad -4.5,\quad -10 $$

On a number line, use a closed circle at $-2$ and shade to the left.

Example 5: Compare absolute values and original numbers

Complete with $<$, $>$, or $=$:

$$ |-7|\ \square\ |-3| $$

Evaluate each absolute value:

$$ |-7|=7,\quad |-3|=3 $$

Now compare:

$$ 7>3 $$

Therefore:

$$ |-7|>|-3| $$

This is different from comparing $-7$ and $-3$, where $-7<-3$.

Example 6: Choose numbers that satisfy an inequality

Give four numbers that satisfy $-1<x\le 3$.

The statement has two conditions:

• $x$ must be greater than $-1$.
• $x$ may be equal to $3$ or less than $3$.

Examples are:

$$ 0,\quad 1,\quad 2.5,\quad 3 $$

The number $-1$ is not allowed, but $3$ is allowed.

Common Mistakes

• Reversing $<$ and $>$. Correction: the open side faces the greater number.
• Thinking $-9>-3$ because $9>3$. Correction: $-9<-3$ on the number line.
• Saying absolute value can be negative. Correction: distance is never negative, so $|-7|=7$.
• Confusing opposite numbers with equal numbers. Correction: $5$ and $-5$ have equal absolute values, but $5\ne -5$.
• Using a closed circle for $x<4$. Correction: use an open circle because $4$ is not included.
• Shading the wrong direction on a number line. Correction: greater than goes right; less than goes left.
• Comparing $|-7|$ and $|-3|$ as if they were $-7$ and $-3$. Correction: evaluate the absolute values first.
• Forgetting that $\le$ and $\ge$ include the endpoint.

Practice Tasks

1. Write the meaning of each symbol: $<$, $>$, $\le$, $\ge$, $\ne$.
2. Complete with $<$ or $>$: $-5\ \square\ -1$.
3. Complete with $<$ or $>$: $\frac{2}{3}\ \square\ 0.7$.
4. Arrange in ascending order: $-2$, $3$, $0$, $-5$, $1$.
5. Evaluate $|-12|$.
6. Evaluate $|8|-|-3|$.
7. Give three numbers that satisfy $x>1$.
8. Give three numbers that satisfy $x\le 0$.
9. Explain the difference between $x<6$ and $x\le 6$.
10. A learner says $|-4|=-4$. Correct the mistake.
11. Complete with $<$, $>$, or $=$: $|-8|\ \square\ |5|$.
12. Give three numbers that satisfy $x\ge -3$.
13. Give three numbers that do not satisfy $x<2$.
14. Describe the number-line diagram for $x>-4$.
15. Complete with $<$ or $>$: $-0.75\ \square\ -0.5$.
16. Arrange in ascending order: $|-2|$, $-4$, $0$, $|3|$, $-1$.
17. Explain why $|a|$ cannot be negative for any number $a$.
18. Correct this statement: "$x\le 5$ is drawn with an open circle at $5$."

Generated Question Layer

• Symbol questions: choose or interpret inequality symbols.
• Ordering questions: compare integers, fractions, decimals, and mixed numbers.
• Number-line questions: represent simple inequalities using open and closed endpoints.
• Absolute-value questions: evaluate expressions with one or more absolute values.
• Explanation questions: describe why a negative number may be less than another negative number.
• Error-analysis questions: repair wrong endpoint marks or incorrect absolute-value signs.

Learner Aid Opportunities

• diagram: number line showing negative numbers, positive numbers, and distance from zero.
• chart: inequality symbols with meanings and examples.
• interactive: open-circle and closed-circle inequality builder.
• animation: movement left and right on the number line to compare values.
• LLM tutor: asks learners to explain comparisons before giving symbolic answers.
• scaffold: side-by-side open-circle and closed-circle number-line templates.
• diagnostic: prompts that separate "compare the numbers" from "compare their absolute values."
• practice generator: mixed integer, decimal, and fraction comparisons with immediate number-line feedback.
• teacher note: quick oral checks using "left of", "right of", "included", and "not included" language.

Exam-Derived Signals

• topic_frequency_2021_2025.json reports $0$ primary mapped records for this topic and includes it in the low-or-no coverage list.
• No direct records for this exact topic were found in question_map_2021_2025.jsonl.
• Related later-topic inequality records appear under topic-inequalities-in-one-unknown and linear programming, but those are separate syllabus topics and should not be merged into this page without review.
• These signals are unreviewed and should be checked against original papers before drawing assessment conclusions.

Source And Review Notes

• Official topic placement comes from data/curriculum_map.json and the syllabus path raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Exam-derived absence is an unreviewed mapping signal, not curriculum authority.
• The page intentionally keeps algebraic inequality solving light and points learners to later pages.
• Learner examples and practice are original prose and need reviewer validation.