Linear Inequalities

Syllabus tag: KCSE | Mathematics | Form 2 | Topic 16 Linear Inequalities

Lesson objectives

By the end of this topic, you should be able to:

• Identify and use inequality symbols.
• Illustrate inequalities on the number line.
• Solve linear inequalities in one unknown.
• Represent linear inequalities graphically in two unknowns.

Linear Inequalities

An inequality states that one quantity is greater or less than another, rather than equal to it.

a) The symbols

< is less than. > is greater than. Neither includes the value itself.

≤ is less than or equal to. ≥ is greater than or equal to. Both include it.

Read from the wide end: it always faces the larger quantity.

b) On the number line

An open circle excludes the value; a filled circle includes it.

The ray points towards all the values that satisfy the inequality.

c) Compound inequalities

A compound inequality traps the unknown between two values.

Read −2 < x ≤ 3 in two halves. First x is greater than −2, and second x is at most 3.

Each end is drawn according to its own symbol, so one may be open and the other filled.

d) Solving

Solve exactly as for an equation. Add, subtract, multiply or divide both sides.

Take 3x + 4 < 19. Subtract 4 to get 3x < 15, then divide by 3 to get x < 5.

e) The one crucial exception

Multiplying or dividing by a negative number reverses the inequality sign.

So from −2x < 6, dividing by −2 gives x > −3, not x < −3.

The check confirms it. Trying x = 0 gives 0 < 6, which is true, and 0 is greater than −3. Trying x = −4 gives 8 < 6, which is false, and −4 is less than −3.

This is the single most examined trap in the topic. Adding and subtracting never reverse the sign; only multiplying or dividing by a negative does.

f) Inequalities in two unknowns

An inequality in two unknowns describes a region of the plane, not a line.

Draw the boundary line first, from the corresponding equation.

Use a solid line for ≤ or ≥, since the boundary is included. Use a dashed line for < or >, since it is not.

Then decide which side to shade.

g) Choosing the side

Pick a test point not on the line. The origin (0, 0) is easiest when the line does not pass through it.

Substitute it into the inequality. If the statement is true, shade that side. If false, shade the other.

Take y ≤ 2x + 1 and test the origin. Since 0 ≤ 1 is true, shade the side containing the origin.

h) Several inequalities together

Draw each boundary and shade each region.

The solution is the area satisfying all of them at once.

Many textbooks shade the unwanted side instead, leaving the required region clear. That is easier to read when three or four inequalities overlap. Say which convention you are using.

i) Where this is used

Budget limits and capacity constraints. Linear programming, where a feasible region is found from several inequalities. Any situation with a maximum or minimum bound.

Words to know

• Inequality -- a statement that one quantity is greater or less than another.
• Strict inequality -- one using > or <, excluding the boundary value.
• Inclusive inequality -- one using ≥ or ≤, including the boundary value.
• Boundary line -- the line drawn from the corresponding equation.
• Feasible region -- the region satisfying all the given inequalities.

:::checkpoint Check yourself

1. Solve 5x − 3 ≥ 12.
2. Solve −3x > 9.
3. Should the boundary line for y < 2x be solid or dashed?
4. Why does dividing by a negative reverse the inequality sign? :::

Bridge to practice

The exercises begin with the symbols and the number line, move through solving and the negative-sign rule, and finish with two-variable regions and word problems. Whenever you divide by a negative, write the reversed sign on a fresh line so the change is visible in your working.