Inequalities

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

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1. Inequalities

Reading the symbols

  • An inequality states that one quantity is greater or less than another rather than equal to it
  • < and > are strict, excluding the end value; ⩽ and ⩾ are inclusive
    • x > 5 excludes 5, whereas x ⩾ 5 includes it

Listing integer solutions

  • Test the two ends independently, since one may be included while the other is not
    • For −2 < x ⩽ 3 the integers are −1, 0, 1, 2 and 3
  • The strict end is excluded and the inclusive end is kept

Worked example

Listing the integers that satisfy an inequality

List the integers n for which −2 ⩽ n < 3.

Solution:

  • Read each end symbol separately, because they are not the same
  • ⩽ on the left means −2 is included
  • < on the right means 3 is not included
  • Working up from −2: −2, −1, 0, 1, 2
  • The next integer would be 3, which the strict sign excludes
  • The integers are −2, −1, 0, 1, 2
  • Zero is an integer and is easy to leave out by accident

2. Inequalities on a number line

The conventions

  • The specification fixes how end points are drawn, and marks are given for using them correctly
  • An open circle marks a strict end point, which is excluded
  • A closed circle marks an inclusive end point, which is included
  • A line joins two end points, and an arrow shows an unbounded direction
A number line in t from −5 to 5 with an open circle at 3 and an arrow running from it off the left-hand end, representing every value below 3 with no lower limit
Source: Solving linear inequalities by Save My Exams
Number line from −5 to 5 with an open circle at −3 joined by a solid line to a closed circle at 4, representing all values greater than −3 and less than or equal to 4
Source: Representing inequalities on a number line by Save My Exams
  • The diagram above represents −3 < x ⩽ 4
A number line from −5 to 5 with a closed circle at −2 joined to an open circle at 1, representing all values from −2 up to but not including 1
Source: Solving linear inequalities by Save My Exams

Exam tip

Join the circles with a line

Drawing an inequality with two ends on a number line is worth two marks, and full marks need both circles right and the line between them. Two correct circles with nothing joining them score only one mark, and so does a correct line with a wrong circle at one end. So check each circle against its own symbol, then join the two with one unbroken line.

Worked example

Drawing an inequality on a number line

Represent each inequality on a number line: (a) −4 < x ⩽ 1, (b) x ⩾ −3.

Solution:

  • Fix each end from its own symbol, then fill in the values in between
  • (a) The left end has <, so −4 is not included: draw an open circle at −4
  • The right end has ⩽, so 1 is included: draw a closed circle at 1
  • x can be any value between them, so join the two circles with a straight line
  • The answer is an open circle at −4 and a closed circle at 1, joined by a line
A number line from −5 to 5 labelled x, with an open circle at −4, a filled circle at 1 and a thick line joining them
  • (b) ⩾ includes −3, so draw a closed circle at −3
  • There is no upper limit, so draw a line from that circle with an arrow pointing to the right
  • The answer is a closed circle at −3 with an arrow to the right
A number line from −5 to 5 labelled x, with a filled circle at −3 and a thick line running from it to an arrowhead at the right-hand end
  • Check: 0 satisfies both, since −4 < 0 ⩽ 1 and 0 ⩾ −3, so both drawings must pass over 0

Worked example

Reading a solution off a number line

A number line has a closed circle at −1 and an open circle at 4, with the line between them shaded. Write down the inequality it represents.

A number line from −3 to 6 labelled x, with a filled circle at −1, an open circle at 4 and a thick line joining them

Solution:

  • Take each end in turn, using the circle to fix the symbol
  • The closed circle at −1 means −1 is included, so that end uses ⩽
  • The open circle at 4 means 4 is excluded, so that end uses <
  • The shading lies between them, so x is between the two values
  • The inequality is −1 ⩽ x < 4
  • Written the other way round it reads 4 > x ⩾ −1, which says the same thing