Inequalities
Section: Algebra and graphs 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Representing and Interpreting Inequalities on a Number Line
An inequality describes a whole range of values rather than a single one. A number line shows this range visually, with the type of circle at each end showing whether that boundary value is included.
- A closed (filled) circle is used for an inclusive inequality (⩽ or ⩾) - the boundary value is included
- An open (unfilled) circle is used for a strict inequality (< or >) - the boundary value is not included
- A compound inequality like -2 x < 3 combines two conditions on the same number line
Worked Example: Listing Integer Solutions
- Question: Find the integer values of x that satisfy -4 < x ⩽ 2.
- Step 1: -4 is excluded (strict <), and 2 is included (inclusive ⩽)
- Step 2: List every integer from just above -4 up to and including 2
- Answer: -3, -2, -1, 0, 1, 2
Common Mistakes
MistakeUsing a closed circle for a strict inequality, or an open circle for an inclusive one
Fixmatch the circle to the symbol: < and > always get an open circle; ⩽ and ⩾ always get a closed circle
MistakeIncluding or excluding a boundary value incorrectly when listing integer solutions, e.g. including -4 in the list for -4 < x ⩽ 2
Fixcheck each end of the inequality separately - a strict inequality excludes that exact value from the list
Solving Linear Inequalities
Extended Only
Solving an inequality uses the same inverse-operation steps as solving an equation - with one extra rule: multiplying or dividing by a negative number flips the direction of the inequality.
- Solve an inequality exactly like an equation, undoing each operation in turn
- If both sides are multiplied or divided by a negative number, the inequality sign must be reversed
- For a compound inequality (three parts), apply the same operation to all three parts at once
Worked Example: Solving a Linear Inequality
- Question: Solve.
5x + 3 > 2x + 18- Step 1: Collect x-terms on one side: 3x > 15
- Step 2: Divide both sides by 3: x > 5
- Answer: x > 5
Worked Example: Dividing by a Negative Number
- Question: Solve.
12 - 4x ⩽ 0- Step 1: Subtract 12 from both sides: -4x -12
- Step 2: Divide both sides by -4, and reverse the inequality: x 3
- Answer: x ⩾ 3
Worked Example: A Compound Inequality
- Question: Solve.
-1 ⩽ 2x + 5 < 13- Step 1: Subtract 5 from all three parts: -6 2x < 8
- Step 2: Divide all three parts by 2: -3 x < 4
- Answer: -3 ⩽ x < 4
Common Mistakes
MistakeForgetting to reverse the inequality sign when multiplying or dividing by a negative number
Fixdividing -4x ⩽ -12 by -4 flips the direction: x ⩾ 3, not x ⩽ 3
MistakeApplying an operation to only one or two parts of a compound inequality
Fixevery operation must be applied to all three parts of a compound inequality at the same time
Graphing Linear Inequalities in Two Variables
Extended Only
An inequality in two variables defines a region of the coordinate grid rather than a single line. The line style shows whether the boundary is included, and shading marks the region that does not satisfy the inequality.
- Draw the boundary line as broken (dashed) for a strict inequality (< or >), or solid for an inclusive inequality (⩽ or ⩾)
- By convention, shade the unwanted region, leaving the region that satisfies the inequality unshaded - unless the question directs otherwise
Worked Example: Drawing and Shading a Region
- Question: On a grid, show the region defined by x ⩾ 2, shading the unwanted region.
- Step 1: Draw x = 2 as a solid vertical line, since ⩾ is inclusive
- Step 2: The wanted region is x ⩾ 2 (to the right of the line), so shade the unwanted region to the left of the line
Common Mistakes
MistakeUsing a solid line for a strict inequality, or a broken line for an inclusive one
Fixmatch the line style to the symbol exactly as with number lines: < and > get a broken line; ⩽ and ⩾ get a solid line
MistakeShading the wanted region instead of the unwanted region, without checking the question's convention
Fixunless told otherwise, shade the region that does not satisfy the inequality, leaving the answer region clear
Listing Inequalities That Define a Region
Extended Only
A shaded region on a graph is usually bounded by several lines at once. Reading off each line's equation, line style, and the side that is shaded gives the full set of inequalities that define the region.
- Find the equation of each boundary line first, using its gradient and intercept
- Check whether each line is solid (⩽ or ⩾) or dashed (< or >)
- Decide whether the shaded region lies above or below each line to choose the correct direction of the inequality
Worked Example: Finding the Inequalities for a Region
- Question: The region marked R is defined by three inequalities: a solid horizontal line at y = 1; a dashed line through (0, -2) and (2, 0); and a solid line through (0, 6) with gradient -1. Region R lies above the horizontal line, above the dashed line, and below the solid sloped line. Find the three inequalities.
- Step 1: The solid horizontal line is y = 1; R lies above it (solid, so inclusive): y 1
- Step 2: The dashed line has gradient (0-(-2)/2-0)=1 and passes through (0,-2), so its equation is y = x - 2; R lies above it (dashed, so strict): y > x - 2
- Step 3: The solid sloped line has gradient -1 and y-intercept 6, so its equation is y = -x + 6; R lies below it (solid, so inclusive): y -x+6
- Answer: y ⩾ 1, y > x - 2, y ⩽ -x + 6
Common Mistakes
MistakeReading solid and dashed lines backwards, treating a solid line as strict and a dashed line as inclusive
Fixa solid line always means the boundary is included (⩽ or ⩾); a dashed line always means it is excluded (< or >)
MistakeGetting the direction of the inequality wrong relative to the shaded side, e.g. writing y < x - 2 when the region actually lies above that line
Fixif the shaded region is above a line, y is greater than that line's expression; if the region is below, y is less than it
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