Learning Resources

Inequalities

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

Grade 06 Pathway: N/A

First Principles

Objective: Represent and solve simple linear inequalities on a number line.

Concrete scenario: Imagine taking a matatu where the minimum fare to town is KSh 50. If you have \(x\) shillings in your pocket, your money must satisfy \(x \ge 50\). Having KSh 50, KSh 70, or KSh 100 works, but KSh 45 is not enough. An inequality describes an entire range of valid solutions rather than a single fixed number.

Geometric insight: We plot these ranges on a number line. A closed dot (\(\bullet\)) means the boundary number is included (for \(\le\) or \(\ge\)). An open dot (\(\circ\)) means the boundary is excluded (for \(<\) or \(>\)). We then shade the line in the direction of all valid solution values.

Algebraic rule: Isolate the variable using standard linear equation steps. Adding or subtracting numbers from both sides preserves the inequality. However, if you multiply or divide both sides by a negative number, you must flip the inequality sign because negative scaling reverses order on the number line.

Open dots for strict inequalities (\(<, >\)); closed dots for non-strict (\(\le, \ge\)).

Shade in the direction the symbol points.

Flip the sign when multiplying or dividing by a negative.

Next lesson: Solving compound inequalities and graphing their intersections.

Key Formulas

\[ax + b < c\] — Isolate \(x\) by subtracting \(b\) and dividing by \(a\). If \(a < 0\), flip the inequality sign.

\[x > k\] — Shade to the right of \(k\). Use an open dot (\(\circ\)) because \(x\) cannot equal \(k\).

\[x \le k\] — Shade to the left of \(k\). Use a closed dot (\(\bullet\)) because \(x\) can equal \(k\).

\[a < x \le b\] — Compound inequality: \(x\) is strictly greater than \(a\) and less than or equal to \(b\). Graph with an open dot at \(a\) and closed dot at \(b\).

Worked Examples

Example 1 (Easy): Solve \(3x - 5 < 7\)

Problem: Solve \(3x - 5 < 7\) for \(x\).

Add 5 to both sides: \(3x < 12\).
Divide by 3: Since 3 is positive, keep the sign unchanged: \(x < 4\).
Graph: Open dot at 4, shaded left.

Answer: \(x < 4\)

Example 2 (Medium): Solve \(-2x + 4 \ge 10\)

Problem: Solve \(-2x + 4 \ge 10\) for \(x\).

Subtract 4 from both sides: \(-2x \ge 6\).
Divide by \(-2\): Because we divide by a negative number, flip \(\ge\) to \(\le\): \(x \le -3\).
Graph: Closed dot at \(-3\), shaded left.

Answer: \(x \le -3\)

Example 3 (Hard): Solve \(4(x - 2) \ge 7x + 7\)

Problem: Solve for \(x\): \(4(x - 2) \ge 7x + 7\).

Expand brackets: \(4x - 8 \ge 7x + 7\).
Collect terms: Subtract \(7x\) from both sides: \(-3x - 8 \ge 7\). Add 8 to both sides: \(-3x \ge 15\).
Divide by \(-3\): Flip the sign to get \(x \le -5\).

Answer: \(x \le -5\)

Common Mistakes

Mistake Graphing \(x > 5\) with a closed dot at 5.

Correction Use an open dot for strict inequalities (\(<, >\)).

Why it feels right 5 is the boundary number, so learners feel it should be colored in. However, \(x > 5\) means 5 itself is not a valid solution.

Mistake Forgetting to flip the sign when dividing by a negative number (e.g., \(-2x > 6 \rightarrow x > -3\)).

Correction \(-2x > 6 \rightarrow x < -3\).

Why it feels right Learners treat inequalities identically to equations. Multiplying or dividing by a negative number reverses numerical order on the number line.

Real World

Pocket Money for Bus Fare: A matatu ride costs at least KSh 50. If you have \(m\) shillings, \(m \ge 50\). Any value from 50 upward is valid.

School Fundraiser Goal: A school club needs to raise strictly more than KSh 2,000 for a trip. If total contributions are \(c\), then \(c > 2000\). Raising exactly KSh 2,000 is not enough.

Vehicle Speed Restrictions: A truck driver must maintain a speed \(s\) greater than 40 km/h but not exceeding 80 km/h: \(40 < s \le 80\).

Practice

Solve for \(x\): \(x + 7 < 12\). (Type only the boundary integer number, e.g., 5)

Review the concepts above.

Solve for \(x\): \(4x \ge 24\). (Type only the boundary integer number, e.g., 6)

Review the concepts above.

Solve for \(x\): \(3x - 4 \le 11\). (Type only the boundary integer value, e.g., 5)

Review the concepts above.

A farmer needs to collect at least 45 eggs today. She has already collected 15 eggs and picks 5 eggs from each remaining nest \(n\). What is the minimum number of nests \(n\) she must visit? (Type only the integer answer, e.g., 6)

Review the concepts above.

Solve for \(x\): \(-5x + 3 \le -22\). (Type only the boundary integer value, e.g., 5)

Review the concepts above.

Solve for \(x\): \(2(3x - 1) > 4x + 8\). (Type only the boundary integer value, e.g., 5)

Review the concepts above.