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 just has an x (and/or a y) etc in it and no x2 terms or terms with higher powers of x
- For example, 3x + 4 ⩾ 7 would be read “3x + 4 is greater than or equal to 7”
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
- If you change the inequality sign to an equals sign you are changing the meaning of the problem
- When you multiply or divide both sides by a negative number, you must flip the sign of the inequality
- e.g. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)
- Never multiply or divide by a variable (x) as this 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 Number Lines and deal with “Double” Inequalities
How do I represent linear inequalities on a number line?
- Inequalities such as and can be represented on a normal number line using an open circle and an arrow
- For , the arrow points to the left of
- For , the arrow points to the right of
- Inequalities such as and can be represented on a normal number line using a solid circle and an arrow
- For , the arrow points to the left of
- For , the arrow points to the right of
- Inequalities such as and can be represented on a normal number line using two circles at and and a line between them
- For or use an open circle
- For or , use a solid circle
- Disjoint inequalities such as " or " can be represented with two circles at and , an arrowed line pointing left from and an arrowed line pointing right from , and a blank space between and
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
Examiner Tip
- Do not change the inequality sign to an equals when solving linear inequalities, you will lose marks in an exam for doing this.
Worked example
(a)
Solve the inequality , illustrating your answer on a number line.
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.
Add 1 to all three parts.
Remember not to change the inequality signs.
Remember not to change the inequality signs.
Divide all three parts by 3.
3 is positive so there is no need to flip the signs.
3 is positive so there is no need to flip the signs.
Illustrate the final answer on a number line, using an open circle at 1 and a closed circle at -2.
(b)
Give your answer to part (a) in set notation
Rewrite your answer using the set notation rules discussed above