Year 11 Exam > Year 11 Notes > Mathematics for GCSE/IGCSE > Solving Linear Inequalities
Solving Linear Inequalities | Mathematics for GCSE/IGCSE - Year 11 PDF Download
| Table of contents |
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| What is a linear inequality? |
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| How do I solve linear inequalities? |
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| How do I represent linear inequalities on a number line? |
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| How do I solve double inequalities? |
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What is a linear inequality?
- An inequality indicates that one expression is greater than (>) or less than (<) another.
- ⩾ means "greater than or equal to," and ⩽ means "less than or equal to."
- A Linear Inequality contains only x (and/or y) without x^2 terms or higher powers of x.
- For instance, 3x - 4 ≥ 7 would be interpreted as "3x - 4 is greater than or equal to 7."
How do I solve linear inequalities?
- Solving linear inequalities follows the same principles as solving linear equations, but you must maintain the inequality sign throughout.
- Changing the inequality sign to an equals sign alters the meaning of the problem.
- When you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. For example,1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)
- Avoid multiplying or dividing by a variable (x) since it could be positive or negative.
- The safest method for rearranging inequalities is to add and subtract terms to consolidate them on one side.
- Understanding how to use number lines and handle "double" inequalities is also crucial.
How do I represent linear inequalities on a number line?
- Inequalities like x < a and x > a can be shown on a regular number line with an open circle at a and an arrow extending to the left for less than and to the right for greater than.
- For inequalities like x ≤ a and x ≥ a, a solid circle at a is used, with an arrow pointing left for less than or equal to and right for greater than or equal to.
- Inequalities such as a < x < b and a ≤ x ≤ b are depicted with two circles at a and b and a line between them.
- For less than or greater than, use open circles.
- For less than or equal to or greater than or equal to, use solid circles.
- Disjoint inequalities like "x < a or x > b" can be shown with two circles at a and b, with an arrowed line extending left from a and right from b, and a gap between a and b.
How do I solve double inequalities?
- Inequalities like a < 2x < b are solved by applying the same operation to all three parts of the inequality.
- This follows the same principles as solving linear inequalities.
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