Linear Inequalities

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 2.0 Algebra | Sub-Strand 2.3 Linear Inequalities (6 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

• Solve linear inequalities in one unknown.
• Represent linear inequalities in one unknown on a number line.
• Form inequalities from real-life statements about limits and minimums.
• Apply inequalities to situations involving budgets, capacity and eligibility.

Linear Inequalities

An equation says two things are equal. An inequality says one is bigger or smaller than the other. Most real limits work this way.

A matatu may carry at most fourteen passengers. A learner needs at least fifty marks to pass. Neither is an exact number, and both are inequalities.

a) The four symbols

There are four to know, and the difference between them matters.

The symbol < means less than, and > means greater than. Neither includes the number itself.

The symbol ≤ means less than or equal to, and ≥ means greater than or equal to. Both include the number itself.

So x < 5 does not allow x to be 5. But x ≤ 5 does.

b) Solving in one unknown

Solve an inequality exactly as you would solve an equation. Do the same thing to both sides until the letter is alone.

The answer is not a single number. It is every number below 5.

c) The one rule that differs

There is a single place where inequalities behave unlike equations.

When you multiply or divide both sides by a negative number, the inequality sign turns around.

Checking with a number from your answer is the surest way to catch this. If the check fails, you have forgotten to flip.

d) Showing the answer on a number line

A number line shows the whole set of answers at once.

An open circle means the number is not included. This is x < 5, so 5 itself is excluded.

A filled circle means the number is included. This is x ≥ 2, so 2 counts.

Getting the circle wrong changes the answer, so decide it from the symbol before you draw.

e) Inequalities in two unknowns

With two letters the answer is no longer part of a line. It is a whole region of the plane.

Draw the boundary line first, as though the inequality were an equation. Then shade the side that satisfies it.

The boundary is dashed for < and >, because those points are not included. It is solid for ≤ and ≥.

To decide which side to shade, test one point. The origin (0, 0) is easiest when the line does not pass through it. If the point works, shade its side.

f) Where this is used

A sacco setting a minimum share contribution is writing an inequality. A lorry with a maximum axle load is limited by one. A farmer working out how many bags will fit in a store uses the same idea.

Words to know

• Inequality -- a mathematical statement comparing two quantities that need not be equal.
• Strict inequality -- one using less than or greater than, where the boundary value is excluded.
• Inclusive inequality -- one using less than or equal to, or greater than or equal to, where the boundary is included.
• Solution set -- the complete range of values that satisfy an inequality.
• Compound inequality -- a statement bounding a quantity from both above and below.

:::checkpoint Check yourself

1. What is the difference between x < 7 and x ≤ 7?
2. Solve 5x − 3 ≥ 12.
3. Solve −3x < 9.
4. When is the boundary line of a region drawn dashed? :::

Bridge to practice

The exercises begin with reading and solving simple inequalities, then test the negative-multiplier rule directly, and finish with translation from worded situations and interpretation of the answer in context. For every question involving a negative coefficient, pause at the moment you divide and ask explicitly whether the sign must flip.