Formula sheet: Linear Inequalities

1.1 The symbols and what they mean

An inequality compares two expressions using one of four signs. The strict signs are less than and greater than; the slack signs are at most and at least. A statement may be numerical, such as 3 less than 5, or literal, such as x less than 5. Two comparisons can be chained into a double inequality, as in 3 less than x less than or equal to 5.

A linear inequality in one variable has the form a x plus b compared with 0, with a not zero. The chapter keeps to linear inequalities in one variable.

1.2 Two rules that solve every linear inequality

Solving an inequality uses the same moves as solving an equation, with one extra caution. Adding or subtracting the same number on both sides changes nothing. Multiplying or dividing both sides by a positive number also changes nothing. But multiplying or dividing by a negative number flips the direction of the sign.

The reason is visible in numbers. We know that 3 is greater than 2, yet minus 3 is less than minus 2. Similarly, minus 8 is less than minus 7, but multiplying both by minus 2 gives 16 greater than 14. The flip is a property of the number line, not a rule to memorise blindly.

A useful habit is to move all variable terms to the side where the coefficient becomes positive. That avoids most negative divisions and removes the commonest source of sign errors.

1.3 Writing the solution set

The answer to an inequality is a set of values, not a single number. When the variable is restricted to natural numbers, the solution set is a finite list or a pattern. When the variable ranges over integers, the set may extend in both directions. When the variable is real, the solution set is an interval.

The default universe for the chapter is the set of real numbers. Only when a question names the natural numbers or the integers does the answer change form, so read that line first every time.

1.4 The number line picture

On a number line, mark the boundary value with a circle and shade the side that satisfies the inequality. An open circle marks a strict inequality, where the endpoint is not part of the solution. A filled circle marks a slack inequality, where the endpoint is included. An arrow at the end of the shaded part shows that it continues without bound.

For a system of two inequalities, draw both solution sets on the same line and take the overlap. The overlap is the region where both statements hold. Writing each inequality as an interval first and intersecting the intervals is often faster than shading by hand.

Graphical questions usually carry two marks: one for the correct endpoint and one for the correct direction of shading. Mark the circle type clearly in the answer.