Inequalities in one unknown

Overview

An inequality compares quantities that are not necessarily equal. Instead of saying two expressions have the same value, it may say that one is greater than, less than, at least, or at most another.

This topic helps learners describe ranges of possible values. In real situations, a value may have a limit rather than one exact answer: a fare must be less than a budget, a mass must not exceed a limit, or a number of items must be at least a required amount.

Unlike many equations, an inequality usually has many answers. For example, $x<5$ is true for $4$, $0$, $-10$, and many other values. The aim is therefore to describe the whole solution set clearly, not just find one value that works.

Prerequisites

• Rational, irrational, and real numbers - Learners should compare positive and negative rational numbers.
• Rational numbers on a number line - Inequality solutions are often shown on number lines.
• Inequalities and absolute values - Earlier number-system work introduces comparison language and absolute value ideas.
• Algebraic expressions and equations - Solving inequalities uses many of the same balance steps as solving equations.
• Linear simultaneous equations - Later algebra may combine more than one condition.

Learning Scope

This chapter covers inequality symbols, reading inequality statements, solving linear inequalities in one unknown, reversing the inequality sign when multiplying or dividing by a negative number, representing solutions on a number line, listing integer solutions, and forming simple inequalities from words.

This page does not fully teach compound inequalities, absolute-value inequalities, inequalities in two unknowns, linear programming, or graphing half-planes. Those topics extend the same comparison idea beyond this Form I page.

Subtopics

Inequality Symbols

The four main inequality symbols are:

• $<$ means less than
• $>$ means greater than
• $\le$ means less than or equal to
• $\ge$ means greater than or equal to

For example, $x < 5$ means $x$ can be any number less than $5$. The value $5$ itself is not included. But $x \le 5$ includes $5$.

Key insight: The line under the symbol means equality is allowed.

Read the symbol from left to right:

• $x<5$ means "$x$ is less than $5$"
• $x>5$ means "$x$ is greater than $5$"
• $5<x$ also means "$5$ is less than $x$", so $x>5$

Misconception note: The "open side" of the symbol points toward the larger quantity. In $3<8$, the open side faces $8$.

Reading And Writing Inequalities

Words can be translated into inequality symbols:

• "less than $10$" becomes $x < 10$
• "more than $3$" becomes $x > 3$
• "at most $12$" becomes $x \le 12$
• "at least $7$" becomes $x \ge 7$

If a bus carries at most $45$ passengers, and $p$ is the number of passengers, then:

$$ p \le 45 $$

Key insight: "At most" and "not more than" use $\le$. "At least" and "not less than" use $\ge$.

Limit words are important:

• "below $20$" usually means $x<20$
• "not above $20$" means $x\le 20$
• "exceeds $20$" means $x>20$
• "minimum of $20$" means $x\ge 20$

Always ask whether the boundary value itself is allowed.

Solving Inequalities Like Equations

Many inequality steps look like equation steps. Add, subtract, multiply, or divide both sides while keeping the statement true.

For example:

$$ x + 4 < 9 $$

Subtract $4$ from both sides:

$$ x < 5 $$

Key insight: The solution is usually a range, not one number.

Solving inequalities uses the same balance idea as equations for addition and subtraction. For example:

$$ \begin{aligned} x - 8 &\ge 3 \\ x - 8 + 8 &\ge 3 + 8 \\ x &\ge 11 \end{aligned} $$

The check is different from checking one equation answer. Try one value inside the range and one value outside the range. For $x\ge 11$, $x=12$ should work and $x=10$ should not.

Multiplying Or Dividing By A Negative Number

When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.

For example:

$$ -2x < 8 $$

Divide both sides by $-2$ and reverse the sign:

$$ x > -4 $$

This rule is necessary because number order reverses when signs change. For example, $2 < 5$, but $-2 > -5$.

Key insight: The sign changes direction only when multiplying or dividing by a negative number, not when adding or subtracting a negative number.

The reversal can be seen on a number line. Since $2<5$, multiplying both sides by $-1$ gives $-2$ and $-5$. But $-2$ is to the right of $-5$, so:

$$ -2>-5 $$

Misconception note: In $7-x<15$, subtracting $7$ does not reverse the sign. The sign reverses later when dividing or multiplying by $-1$.

Number-Line Representation

Solutions can be shown on a number line. Use an open circle when the endpoint is not included, and a filled circle when it is included.

For $x > 2$, use an open circle at $2$ and shade to the right. For $x \le -1$, use a filled circle at $-1$ and shade to the left.

Key insight: Greater-than solutions go to the right on the number line; less-than solutions go to the left.

Use this number-line routine:

1. Mark the endpoint.
2. Decide whether the circle is open or filled.
3. Choose the shading direction by testing an easy number.

For $x>-2$, the endpoint is $-2$, the circle is open, and values to the right such as $0$ are shaded because $0>-2$.

Listing Integer Solutions

Some questions ask for integer values satisfying an inequality. Integers are whole numbers and their negatives, such as $\ldots, -2, -1, 0, 1, 2, \ldots$.

If:

$$ x > -3 $$

then the first few integer solutions are:

$$ -2,\ -1,\ 0,\ 1,\ 2,\ldots $$

If:

$$ x \le 4 $$

then the greatest integer solution is $4$.

Key insight: Listing integer solutions depends on whether the endpoint is included.

When the inequality has two boundaries, check both ends. For:

$$ -2 \le x < 4 $$

the integer solutions are:

$$ -2,\ -1,\ 0,\ 1,\ 2,\ 3 $$

The $-2$ is included because of $\le$, but $4$ is not included because of $<$.

Forming Inequalities From Context

Real situations often use limit words. If a learner has $5,000$ shillings and each notebook costs $800$ shillings, the number of notebooks $n$ they can buy must satisfy:

$$ 800n \le 5,000 $$

Solving gives:

$$ n \le 6.25 $$

Since notebooks are counted as whole items, the learner can buy at most $6$ notebooks.

Key insight: After solving, interpret the answer in the context. Some quantities must be whole numbers.

Context can change the final wording. If $n\le 6.25$ and $n$ counts notebooks, then $n$ cannot be $6.25$. The greatest possible whole number is $6$. But if $d\le 6.25$ measures distance in kilometres, $6.25$ km may be a sensible answer.

Key Terms

• Inequality: A mathematical statement comparing quantities using $<$, $>$, $\le$, or $\ge$.
• Unknown: A value represented by a letter.
• Solution set: All values that make an inequality true.
• Endpoint: The boundary value in an inequality, such as $3$ in $x \ge 3$.
• Open circle: A number-line mark showing that an endpoint is not included.
• Filled circle: A number-line mark showing that an endpoint is included.
• Integer solution: A solution that is an integer.
• At most: Less than or equal to.
• At least: Greater than or equal to.
• Reverse the sign: Change $<$ to $>$ or $\le$ to $\ge$, usually after multiplying or dividing by a negative number.

Worked Examples

Example 1: Solve A Simple Inequality

Solve $x + 6 \le 14$.

$$ \begin{aligned} x + 6 &\le 14 \\ x &\le 8 \end{aligned} $$

The solution is $x \le 8$.

Example 2: Solve With Brackets

Solve $3(x - 2) > 12$.

$$ \begin{aligned} 3(x - 2) &> 12 \\ 3x - 6 &> 12 \\ 3x &> 18 \\ x &> 6 \end{aligned} $$

The solution is $x > 6$.

Example 3: Divide By A Negative Number

Solve $10 - x \le 3$.

$$ \begin{aligned} 10 - x &\le 3 \\ -x &\le -7 \\ x &\ge 7 \end{aligned} $$

The solution is $x \ge 7$. The inequality sign reverses when both sides are multiplied by $-1$.

Example 4: List Integer Values

Solve $10 - x \le 3(x + 10)$ and state the first four integer values satisfying the inequality.

$$ \begin{aligned} 10 - x &\le 3(x + 10) \\ 10 - x &\le 3x + 30 \\ 10 - 30 &\le 3x + x \\ -20 &\le 4x \\ -5 &\le x \end{aligned} $$

So:

$$ x \ge -5 $$

The first four integer values satisfying the inequality are:

$$ -5,\ -4,\ -3,\ -2 $$

Example 5: Context Inequality

A small bag can carry at most $18 \text{ kg}$. One book has mass $1.2 \text{ kg}$. If $b$ books are placed in the bag, find the possible values of $b$.

Form the inequality:

$$ 1.2b \le 18 $$

Solve:

$$ \begin{aligned} 1.2b &\le 18 \\ b &\le 15 \end{aligned} $$

Since $b$ counts books, $b$ can be $0, 1, 2, \ldots, 15$.

Example 6: Variables On Both Sides

Solve $5x - 4 > 2x + 8$.

Collect the variable terms on the left and the number terms on the right:

$$ \begin{aligned} 5x - 4 &> 2x + 8 \\ 5x - 2x &> 8 + 4 \\ 3x &> 12 \\ x &> 4 \end{aligned} $$

The solution is $x>4$. No sign reversal is needed because the final division is by positive $3$.

Example 7: Negative Coefficient With Brackets

Solve $-2(x-3)\le 10$.

First expand carefully:

$$ \begin{aligned} -2(x-3) &\le 10 \\ -2x+6 &\le 10 \\ -2x &\le 4 \end{aligned} $$

Now divide by $-2$ and reverse the sign:

$$ x \ge -2 $$

Check with a value inside the solution set, such as $x=0$:

$$ -2(0-3)=6,\quad 6\le 10 $$

So the answer is reasonable.

Example 8: Integer Solutions In A Finite Range

List all integer values satisfying $-1 < x \le 3$.

The lower endpoint $-1$ is not included, but the upper endpoint $3$ is included. The integer values are:

$$ 0,\ 1,\ 2,\ 3 $$

Common Mistakes

• Reversing the sign after adding or subtracting: The sign reverses only when multiplying or dividing by a negative number.
• Forgetting to reverse the sign after dividing by a negative coefficient.
• Treating $\le$ as $<$: If equality is allowed, the endpoint belongs to the solution.
• Listing integers from the wrong side of the number line.
• Giving one answer when the solution is a range.
• Ignoring context: A solution such as $n \le 6.25$ may mean $n \le 6$ when $n$ counts whole objects.
• Misreading "at most" as $\ge$ or "at least" as $\le$.

Practice Tasks

Foundation

1. Write the meaning of $x < 9$ in words.
2. Translate "the number of pupils is at least $40$" into an inequality.
3. Translate "the mass is not more than $15$ kg" into an inequality.
4. State whether the endpoint is included in $x\le -3$.

Solve And Represent

1. Solve $x - 5 > 11$.
2. Solve $2x + 3 \le 17$.
3. Solve $4(x + 1) > 20$.
4. Solve $-3x \ge 12$.
5. Solve $7 - 2x < 15$.
6. Solve $6x+1\le 2x+17$.
7. Represent $x \ge -2$ on a number line.
8. Represent $x<4$ on a number line.

Integer And Context Practice

1. List the first five integer solutions of $x > 3$.
2. List all integer solutions of $-2 \le x < 4$.
3. List all integer solutions of $-5<x\le 1$.
4. A taxi fare must be less than or equal to $12,000$ shillings. If each kilometre costs $1,500$ shillings, form and solve an inequality for the number of kilometres.
5. A container can hold at most $25 \text{ litres}$. Each bottle contains $1.5 \text{ litres}$. Find the greatest whole number of bottles that can be poured into the container.
6. A school needs at least $240$ exercise books. Each box contains $35$ books. Form an inequality and find the least whole number of boxes required.

Generated Question Layer

• Direct understanding questions: Ask learners to match symbols with phrases such as "at least" and "not more than".
• Skill questions: Generate one-step, two-step, bracket, and negative-coefficient inequalities.
• Number-line questions: Ask learners to choose open or filled circles and the correct shading direction.
• Integer-solution questions: Ask for first values, greatest values, least values, or all values in a finite interval.
• Application problems: Use capacity, money, distance, age, mass, classroom limits, and production limits.
• Edge cases: Include division by negative numbers, endpoint inclusion, non-integer contextual answers, and inequalities with variables on both sides.
• LLM tutor: Prompt the tutor to ask whether the endpoint is included before accepting a number-line or integer-list answer.

Learner Aid Opportunities

• diagram: Number-line diagrams showing open and filled endpoints.
• chart: Phrase-to-symbol chart for "less than", "at most", "not more than", "greater than", "at least", and "not less than".
• graph: Simple one-dimensional graph view of solution intervals with endpoint inclusion shown by open and filled circles.
• animation: Inequality sign flipping when both sides are multiplied by a negative number, using $2<5$ becoming $-2>-5$.
• interactive: Number-line tool where learners solve an inequality and drag the correct endpoint, circle type, and shading direction.
• interactive: Integer-solution selector where learners click all integers in a finite interval and receive endpoint feedback.
• video: Short explanation of why dividing by a negative reverses order, followed by one bracket example.
• LLM tutor: Step checker that asks, "Did you multiply or divide by a negative number in this step?" and "Is the endpoint included?"

Exam-Derived Signals

The first automatic 2021-2025 Paper 1 mapping counted 2 total Inequalities in one unknown records: 1 in 2021 and 1 in 2023. These records are unreviewed extraction signals, not verified official past-question links.

The 2022 examination format crosswalk maps the format group Algebra/Quadratic equations to this topic, together with algebraic expressions, quadratic equations, and simultaneous equations. The crosswalk record is marked official, but it gives a broad group-level signal rather than a separate weight for inequalities.

Recent unreviewed extracted signals include:

| Year | Question ID | Signal | | ---: | --- | --- | | 2021 | csee_041_2021_p1_q14_a_i | Writing inequalities from a production context; mapped with a multi-topic review flag. | | 2023 | csee_041_2023_p1_q10_b | Solving a linear inequality and listing the first four integer values. |

A 2025 extracted record also maps inequality language as a secondary signal inside a linear-programming formulation. That record belongs primarily to Linear programming and should not be used as a reviewed Form I inequality example without manual checking.

Source And Review Notes

• Official syllabus status: The topic identity, form placement, competence, source topic ID, and hub come from the 2023 CSEE Mathematics syllabus through data/curriculum_map.json.
• Official source reference: The cited syllabus file is raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Exam signal status: The 2021-2025 signals come from data/topic_frequency_2021_2025.json and data/question_map_2021_2025.jsonl; they are unreviewed and should not be treated as audited past-question references.
• Exam format status: The 2022 format crosswalk maps Algebra/Quadratic equations to this page as one member of a broader algebra group.
• Content authorship status: Explanations, worked examples, and practice tasks are original learner-facing prose written from the syllabus topic and assessment signals, not copied from textbooks, Wikipedia, or extracted solutions.
• Renderer QA: This page uses $...$ and $$...$$ math notation for compatibility with Obsidian, KaTeX, and MathJax. Some plain Markdown viewers may show the raw delimiters.