Solving Linear Inequalities
What is a linear inequality?
- An inequality tells you that one expression is greater than (“>”) or less than (“<”) another
- “⩾” means “greater than or equal to”
- “⩽” means “less than or equal to”
- A linear inequality only has constant terms (numbers with no letters) and terms in x (and/or y)
- but no x2 terms or terms with other powers of x
- 3x2 > 12 is not a linear inequality (it is a quadratic inequality)
- but no x2 terms or terms with other powers of x
How do I solve linear inequalities?
- Solving linear inequalities is just like solving linear equations
- Follow the same rules, but keep the inequality sign throughout
- Changing the inequality sign to an equals sign changes the meaning of the problem
- Follow the same rules, but keep the inequality sign throughout
- When you multiply or divide both sides by a negative number, you must flip the sign of the inequality
- e.g. 1 < 2
- Multiply both sides by –1 (negative number)
- It becomes –1 > –2 (sign flips)
- e.g. 1 < 2
- Never multiply or divide by a variable (x)
- It could be positive or negative
- The safest way to rearrange is simply to add & subtract to move all the terms onto one side
- You also need to know how to
- use set notation
- deal with “double” inequalities
How do I represent linear inequalities using set notation?
- We use curly brackets and a colon in set notation
- means "x is in the set such that ..."
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- e.g. the set of all x such that x is greater than 3 is written
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- If x is between two values
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- you can write it as a single set
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- e.g. if x is greater than 3 and less than or equal to 5, then in set notation
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- or you can write it as separate sets, using the intersection symbol,
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- so the above example could also be written in set notation as
- Either way will get the marks, unless a question specifically asks for one or the other
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- If x is less than one value OR greater than another value (disjoint sets)
- then the two end values must be written in separate sets using the union symbol,
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- e.g. if x is less than 3 or greater than or equal to 5, then in set notation
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How do I solve double inequalities?
- Inequalities such as can be solved by doing the same thing to all three parts of the inequality
- Use the same rules as solving linear inequalities
Exam Tip
- Do not change the inequality sign to an equals when solving linear inequalities
- You will lose marks in an exam for doing this.
- Remember to reverse the direction of the inequality sign when multiplying or dividing by a negative number!
Worked example
This is a double inequality, so any operation carried out to one side must be done to all three parts.
Use the expression in the middle to choose the inverse operations needed to isolate x.
Remember not to change the inequality signs.
3 is positive so there is no need to flip the signs.
Rewrite your answer using the set notation rules discussed above
Worked example
Solve the inequality .
Subtract 5 from both sides, keeping the inequality sign the same
Now divide both sides by -2.
However because you are dividing by a negative number, you must flip the inequality sign
The final answer is normally written with the number first, but you won't be penalised for writing the x first so long as the inequality sign is the correct way around
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