Learning Resources

Linear Inequalities

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 07 Pathway: N/A

First Principles

Objective

Understand the foundational meaning of linear inequalities in one variable, solve them systematically, and represent their solution sets accurately on a number line.

Core Analogy: The Tilting Scale

Unlike an equation which represents a balanced scale at a single point (\(=\)), an inequality represents a scale tilted to one side. A solution to an inequality is not just one number, but an entire continuous region of numbers that keep the scale tilted correctly.

When you multiply or divide both sides by a negative number, the physical direction of comparison reverses completely—just like flipping the scale upside down—so the inequality symbol must reverse its direction.

(a) Real-World Boundary: The Matatu Capacity Rule

In Kenya, traffic laws state that a standard 14-seater matatu can carry at most 14 passengers. If \(p\) represents the number of passengers, the law states:

\[p \le 14\]

If there are already 4 passengers on board and groups of size 2 board at subsequent stages, the inequality \(4 + 2x \le 14\) describes all valid scenarios.

(b) Geometric Representation: Rays and Circles

We graph solution sets on a horizontal number line:

Strict Inequalities (\(<\) or \(>\)): Represented with an open circle (\(\circ\)) at the boundary point to show that the boundary itself is excluded.
Inclusive Inequalities (\(\le\) or \(\ge\)): Represented with a closed/filled circle (\(\bullet\)) at the boundary point to show that the boundary itself is included.
Shading: A solid thick ray is drawn in the direction of all valid solutions (pointing left for \(<, \le\) and right for \(>, \ge\)).

Key Formulas

\[ax + b < c \quad \Longleftrightarrow \quad ax < c - b \quad \Longleftrightarrow \quad x < \frac{c - b}{a} \quad (\text{when } a > 0)\] — Standard transformation: isolate the variable term, then divide by the coefficient while preserving inequality direction.

\[-ax + b \le c \quad \Longleftrightarrow \quad -ax \le c - b \quad \Longleftrightarrow \quad x \ge \frac{c - b}{-a} \quad (\text{when dividing by } -a < 0)\] — The Golden Rule: Multiplying or dividing both sides by a negative value reverses the inequality sign.

\[p \le ax + b < q \quad \Longleftrightarrow \quad p - b \le ax < q - b \quad \Longleftrightarrow \quad \frac{p - b}{a} \le x < \frac{q - b}{a}\] — Compound Inequalities: Perform identical algebraic operations on all three parts simultaneously.

\[\begin{aligned} x > k &\implies (k, \infty) \quad [\text{Open circle } \circ \text{ at } k, \text{ arrow right}] \\ x \le k &\implies (-\infty, k] \quad [\text{Closed circle } \bullet \text{ at } k, \text{ arrow left}] \end{aligned}\] — Graphical and interval notation conventions.

Worked Examples

Problem 1 (Easy): Solve the linear inequality \(3x - 5 < 7\) and represent the solution on a number line.

Isolate the variable term: Add \(5\) to both sides. \[3x - 5 + 5 < 7 + 5\] \[3x < 12\]
Solve for \(x\): Divide both sides by \(3\). Since \(3 > 0\), the sign remains unchanged. \[\frac{3x}{3} < \frac{12}{3} \implies x < 4\]
Graph on Number Line: Place an open circle at \(4\) (since \(4\) is excluded) and draw a ray shading to the left toward \(-\infty\).

Final Answer: \(x < 4\) or in interval notation \((-\infty, 4)\).

Problem 2 (Medium): Solve the inequality \(5 - 2x \ge 11\) and state the largest integer that satisfies the inequality.

Isolate the variable term: Subtract \(5\) from both sides. \[5 - 2x - 5 \ge 11 - 5\] \[-2x \ge 6\]
Divide by the coefficient of \(x\): Since \(-2\) is negative, reverse the inequality sign from \(\ge\) to \(\le\). \[x \le \frac{6}{-2} \implies x \le -3\]
Identify largest integer: Since \(x \le -3\), the set of integer solutions is \(\{\dots, -5, -4, -3\}\). The maximum integer is \(-3\).

Final Answer: \(x \le -3\); the largest integer is \(-3\).

Problem 3 (Hard): Solve the compound inequality \(-7 < 3 - 2x \le 9\) and determine the set of all integer values of \(x\).

Subtract \(3\) from all three parts: \[-7 - 3 < 3 - 2x - 3 \le 9 - 3\] \[-10 < -2x \le 6\]
Divide all three parts by \(-2\): Because \(-2 < 0\), reverse both inequality signs. \[\frac{-10}{-2} > x \ge \frac{6}{-2}\] \[5 > x \ge -3\]
Rewrite in standard order (least to greatest): \[-3 \le x < 5\]
List integer solutions: \(x \in \{-3, -2, -1, 0, 1, 2, 3, 4\}\).

Final Answer: \(-3 \le x < 5\) (interval: \([-3, 5)\)). Total of 8 integers.

Common Mistakes

Mistake 1: Forgetting to reverse the inequality sign when dividing by a negative number.
Example: Writing \(-2x \ge 6 \implies x \ge -3\).

Correction \(-2x \ge 6 \implies x \le -3\).

Why it feels right In standard linear equations (e.g., \(-2x = 6\)), the equality sign never flips. Students carry this automatic habit over to inequalities. But testing \(x = 0\) quickly reveals \(-2(0) = 0 \not\ge 6\), confirming the sign must flip.

Mistake 2: Confusing open (\(\circ\)) and closed (\(\bullet\)) circles on the number line.
Example: Drawing a closed circle for \(x > 4\).

Correction An open circle \(\circ\) means the boundary point is excluded (\(<, >\)). A closed circle \(\bullet\) means the boundary point is included (\(\le, \ge\)).

Why it feels right Learners often mark the number itself because it appears in the written answer, forgetting that strict inequality excludes the boundary value.

Mistake 3: Flipping the inequality sign when subtracting a number.
Example: Changing \(x + 5 < 10\) to \(x > 5\) after subtracting 5.

Correction Addition and subtraction never flip the inequality sign. Only multiplying or dividing by a negative number flips it.

Real World

Real-World Applications in Kenya

Inequalities govern real-life thresholds, constraints, and budgets where exact equality is neither required nor realistic.

1. Matatu Transport Capacity & Revenue

A matatu conductor in Nairobi charges Ksh 50 per passenger. The daily vehicle lease and fuel cost total Ksh 3,200. To turn a profit, daily earnings must exceed costs:

\[50n > 3200 \implies n > 64\]

The crew must transport at least 65 passengers each day to make a profit.

2. Maize Storage & Moisture Thresholds

In the North Rift region (Eldoret), the National Cereals and Produce Board (NCPB) specifies that the moisture content \(m\) of harvested maize delivered for long-term storage must not exceed \(13.5\%\):

\[m \le 13.5\%\]

If drying reduces moisture by \(1.2\%\) per sun-drying session from an initial \(18.3\%\), the inequality is \(18.3 - 1.2s \le 13.5\). Solving gives \(-1.2s \le -4.8 \implies s \ge 4\) sessions.

3. Mobile Money (M-Pesa) Daily Limits

A small kiosk merchant can hold a maximum wallet balance of Ksh 500,000. If their opening balance is Ksh 120,000 and they receive daily business transactions averaging \(x\) shillings over 5 days, the business constraint is \(120000 + 5x \le 500000\).

Practice

A market vendor in Gikomba sells umbrellas for Ksh 150 each. He wants his total sales to be at least Ksh 2,100 in one day. What is the minimum whole number of umbrellas he must sell? (Type only the number, e.g., 42)

Review the concepts above.

A farmer in Nakuru has 180 metres of land boundary available to plant rows of maize. If each row requires at least 15 metres of spacing along the boundary, what is the greatest number of rows the farmer can plant? (Type only the number, e.g., 42)

Review the concepts above.

A matatu operator charges a fixed base fee of Ksh 20 plus Ksh 5 per kilometre travelled. If a passenger's total fare must not exceed Ksh 120, what is the maximum distance in whole kilometres the passenger can travel? (Type only the number, e.g., 42)

Review the concepts above.

A student has a travel budget of Ksh 500. A registration card fee costs Ksh 10 once, and each subsequent bus ride costs Ksh 45. What is the maximum number of whole bus rides the student can afford without exceeding the budget? (Type only the number, e.g., 42)

Review the concepts above.

A fruit vendor pays a fixed daily market stall fee of Ksh 500. She earns a profit of Ksh 150 on each crate of mangoes sold. If she aims to make a net daily profit of at least Ksh 2,000 (after paying the stall fee), what is the minimum whole number of crates she must sell? (Type only the number, e.g., 42)

Review the concepts above.

A builder requires strictly more than 119 stone blocks for a foundation. An initial pile of 49 blocks is on site, and additional blocks arrive in bundles of 7. What is the smallest whole number of bundles (x) needed to exceed 119 blocks? (Type only the number, e.g., 42)

Review the concepts above.