Inequalities in IGCSE Maths
Solving linear inequalities, showing solutions on a number line, and the Extended skill of graphical inequalities: drawing regions and finding the inequalities that define a region. Clear diagrams, worked examples and an exam-style question, set out the way the mark scheme wants.
0580 · 2.6Inequality symbols
An inequality shows that one quantity is bigger or smaller than another, rather than exactly equal:
\(<\) less than
\(>\) greater than
\(\le\) less than or equal to
\(\ge\) greater than or equal to
For example, \(x \ge 5\) means \(x\) can be \(5\) or any number bigger than \(5\).
Two inequalities can be combined into one. \(-2 < x \le 3\) means \(x\) is greater than \(-2\) and less than or equal to \(3\). That is every value in between.
Solving linear inequalities
Solve a linear inequality exactly as you would the matching equation, doing the same thing to both sides, with one extra rule to watch for.
If you multiply or divide both sides by a negative number, you must reverse the inequality sign. Adding or subtracting does not flip it. Only multiplying or dividing by a negative does.
Solve \(3x - 4 \le 11\).
Solve \(7 - 2x < 1\).
Subtract \(7\) from both sides:
\[-2x < -6\]Divide both sides by \(-2\) and reverse the sign:
\[x > 3\]Inequalities on a number line
We show the solution of an inequality on a number line, using a circle at the boundary value and an arrow for the direction of all the values that work:
Empty circle - strict (\(<\) or \(>\)) - does not include the end value
Filled circle - inclusive (\(\le\) or \(\ge\)) - includes the end value
This shows \(x > 2\): an empty circle at \(2\) and an arrow pointing right.
Listing integer solutions
A common exam task is to list the integers that satisfy a compound inequality. The circles tell you whether each end value is included.
List the integers that satisfy \(-1 < x \le 3\).
The empty circle excludes \(-1\); the filled circle includes \(3\):
\[x = 0,\ 1,\ 2,\ 3\]Graphical inequalities
ExtendedOn a coordinate grid, a linear inequality describes a region, meaning all the points on one side of a boundary line. To draw it: replace the inequality sign with \(=\) to get the boundary line, draw it, then use a test point such as the origin to decide which side you want.
Use a broken (dashed) line for a strict inequality (\(<\), \(>\)) and a solid line for an inclusive one (\(\le\), \(\ge\)). In Cambridge IGCSE you then shade the region you do not want, leaving the required region (labelled \(R\)) clear. This is the opposite of many textbooks, so read the instruction carefully.
Show the region that satisfies \(y < x + 2\).
Draw \(y = x + 2\) as a broken line, since the inequality is strict. Test \((0,0)\): \(0 < 0 + 2\) is true, so the origin is in the region we want. Following the convention, shade the other side, the unwanted region above the line, leaving \(R\) clear.
Finding inequalities from a region
ExtendedIf a region is already drawn, you can write down the inequalities that define it. For each boundary line, find its equation, then use a test point inside the region to decide which way the sign goes. A solid line gives \(\le\) or \(\ge\); a broken line gives \(<\) or \(>\).
Write down the three inequalities that define region \(R\).
The boundary lines are \(x = 1\), \(y = 1\) and \(x + y = 5\), all solid. Test a point inside \(R\), say \((2,2)\): it is to the right of \(x = 1\), above \(y = 1\), and below \(x + y = 5\). So:
\[x \ge 1, \quad y \ge 1, \quad x + y \le 5\]Exam-style question
ExtendedThe region \(R\) is shown on the grid below, bounded by three lines.
(a) Write down the three inequalities that define region \(R\). [3]
(b) Write down the coordinates of the point in \(R\), with integer coordinates, that has the largest value of \(x + y\). [2]
(c) How many points in \(R\) have integer coordinates with \(y = 2\)? [1]
Show solution (a)
The boundaries are \(y = 1\), \(x = 4\) and \(y = x + 1\), all solid. Testing \((3, 2)\) inside \(R\) gives:
\[y \ge 1, \quad x \le 4, \quad y \le x + 1\]Show solution (b)
The largest \(x + y\) sits at the top-right corner of \(R\), where \(x = 4\) meets \(y = x + 1\):
\[(4, 5), \quad x + y = 9\]Show solution (c)
With \(y = 2\): the conditions \(y \le x + 1\) and \(x \le 4\) give \(1 \le x \le 4\), so \(x = 1, 2, 3, 4\):
\[4 \text{ points}\]Region questions appear most often on Paper 4. The marks are won by getting the small details right: a broken line for a strict inequality, a solid line for an inclusive one, and shading the unwanted side. Always confirm each inequality with a test point inside \(R\) before writing it down.
Common mistakes
The inequality sign reverses only when you multiply or divide both sides by a negative number. Watch for this whenever the variable ends up with a negative coefficient.
A strict inequality (\(<\), \(>\)) uses an empty circle; an inclusive one (\(\le\), \(\ge\)) uses a filled circle. Getting these the wrong way round loses easy marks.
On a graph, a strict inequality is a broken line and an inclusive one is a solid line. This is the same idea as empty and filled circles, applied to the boundary.
Cambridge asks you to shade the region you do not want, leaving \(R\) clear. Always test a point and read the instruction before you shade.
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