Algebra
Inequalities
Year 10 · Year 11
- ✓Solve linear inequalities in one variable and represent the solution set on a number line.
- ✓Solve linear inequalities in two variables and represent the solution set graphically.
- ✓Identify the region that satisfies a set of linear inequalities on a graph.
- ✓Solve quadratic inequalities in one variable by finding critical values and sketching the graph (Higher tier).
- ✓Represent the solution set of a quadratic inequality on a number line.
Key concepts
An inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations which show equality, inequalities show that one expression is greater than, less than, greater than or equal to, or less than or equal to another. The symbols used are: '<' (less than), '>' (greater than), '≤' (less than or equal to), '≥' (greater than or equal to).
Solving linear inequalities is very similar to solving linear equations. You can add or subtract the same value from both sides, and multiply or divide both sides by a positive value without changing the inequality sign. However, if you multiply or divide both sides by a negative value, you MUST reverse the direction of the inequality sign.
The solution set of an inequality in one variable can be shown on a number line. Use an open circle (or hollow circle) for strict inequalities (< or >) to show that the endpoint is NOT included in the solution. Use a closed circle (or solid circle) for inclusive inequalities (≤ or ≥) to show that the endpoint IS included in the solution. Draw an arrow from the circle in the direction of the solution.
Inequalities involving two variables (e.g., x and y) can be represented as regions on a coordinate plane. First, draw the boundary line corresponding to the equality (e.g., for y > 2x + 1, draw y = 2x + 1). Use a solid line for inclusive inequalities (≤ or ≥) to show that points on the line ARE part of the solution. Use a dashed line for strict inequalities (< or >) to show that points on the line are NOT part of the solution. To determine the solution region, test a point (e.g., (0,0) if it's not on the line) in the original inequality. If the test point satisfies the inequality, then that region is the solution. In UK exams, it is common practice to shade the region that does NOT satisfy the inequality (the unwanted region), leaving the solution region unshaded.
Solving quadratic inequalities involves finding the range of values for x that satisfy the inequality. This typically involves: 1. Rearranging the inequality so one side is zero (e.g., ax² + bx + c > 0). 2. Finding the critical values by solving the corresponding quadratic equation (ax² + bx + c = 0). These are the roots of the quadratic. 3. Sketching the graph of the quadratic function (y = ax² + bx + c). The shape of the parabola (U-shape for a > 0, n-shape for a < 0) and its x-intercepts (the critical values) are key. 4. Using the sketch to identify the regions where the graph satisfies the inequality (e.g., where y > 0 or y < 0).
Key facts to remember
- 1The inequality symbols are: '<' (less than), '>' (greater than), '≤' (less than or equal to), '≥' (greater than or equal to).
- 2When multiplying or dividing both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.
- 3On a number line, an open circle indicates the endpoint is not included (< or >), and a closed circle indicates it is included (≤ or ≥).
- 4For graphical inequalities, use a dashed line for strict inequalities (< or >) and a solid line for inclusive inequalities (≤ or ≥).
- 5In UK exams, for graphical inequalities, it is standard practice to shade the region that does NOT satisfy the inequality (the unwanted region).
- 6To solve quadratic inequalities, find the critical values (roots), sketch the graph of the quadratic, and use the sketch to determine the solution regions.
- 7A quadratic inequality will often have two separate solution regions (e.g., x < a or x > b) or a single region (e.g., a < x < b).
Worked examples
Example 1
Solve 3x - 5 < x + 7 and represent the solution on a number line.
Answer
x < 6
Remember to use an open circle for strict inequalities (< or >).
Example 2
On a coordinate grid, show the region that satisfies y ≥ 2x - 1.
Answer
A coordinate grid with a solid line y = 2x - 1 passing through (0, -1) and (2, 3). The region below the line is shaded, leaving the region above and including the line unshaded.
Always use a solid line for 'greater than or or equal to' or 'less than or equal to'. Shade the unwanted region.
Example 3
Solve x² - 5x + 6 > 0. (Higher Tier)
Answer
x < 2 or x > 3
Always sketch the quadratic graph to correctly identify the regions for the solution.
Common mistakes
- ✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
- ✗Using the wrong type of circle (open/closed) when representing inequalities on a number line.
- ✗Using the wrong type of line (solid/dashed) when drawing boundary lines for graphical inequalities.
- ✗Shading the wanted region instead of the unwanted region for graphical inequalities (always check exam board specific instructions, but unwanted is standard).
- ✗For quadratic inequalities, simply finding the roots and not considering the shape of the parabola to determine the correct regions.
- ✗Writing 'a < x > b' instead of 'x < a or x > b' for quadratic inequalities with two separate regions.
Exam tips
- ★Always show full working for solving inequalities, especially when rearranging terms or reversing signs.
- ★For graphical inequalities, use a ruler to draw straight lines accurately and label your lines clearly.
- ★When solving quadratic inequalities, a quick sketch of the parabola is invaluable for determining the correct solution regions.
- ★Double-check your answer by substituting a value from your solution set (and one not in it) back into the original inequality.
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