Inequalities
Every question with a fully worked solution.
- Inequalities - Video - How To Solve Linear Inequalities, Basic Introduction Watch
Theory
A linear inequality such as \(3x - 1 < 11\) has a whole range of solutions, not just one. This Year 9 maths page shows how to solve inequalities, graph the solution on a number line, and when to reverse the inequality sign.
An inequality compares two quantities that may not be equal. The signs are \(<\) (less than), \(>\) (greater than), \(\le\) (less than or equal to) and \(\ge\) (greater than or equal to).
The solution of an inequality is every number that makes it true. \(x > 4\) includes \(4.1\), \(5\), \(100\) and infinitely many other numbers, but not \(4\) itself.
Show a solution on a number line. An open circle means the end number is not included (\(<\) or \(>\)). A closed circle means it is included (\(\le\) or \(\ge\)). Shade the line in the direction of the solutions.
Solve an inequality like an equation, with one extra rule: when you multiply or divide both sides by a negative number, reverse the inequality sign. Adding, subtracting, and multiplying or dividing by a positive number do not change the sign.
For any numbers \(a\) and \(b\), and a positive number \(k\):
| Inequality | In words | Circle | Shade |
|---|---|---|---|
| \(x > a\) | greater than \(a\) | open | right |
| \(x \ge a\) | greater than or equal to \(a\) | closed | right |
| \(x < a\) | less than \(a\) | open | left |
| \(x \le a\) | less than or equal to \(a\) | closed | left |
How to solve a linear inequality
- Clear fractions and expand brackets, as for an equation.
- Collect the pronumeral terms on one side and the numbers on the other.
- Divide by the coefficient. If it is negative, reverse the sign.
- Read the answer with the pronumeral first: \(-3 < x\) is \(x > -3\).
- Check with one number inside the solution and one outside.
Whole-number answers
- Greatest whole number: from \(n \le 38.53\ldots\), round down to \(38\).
- Smallest whole number: from \(4n + 3 > 25\) you get \(n > 5.5\), so the answer is \(6\). With \(n > 6\) exactly, the answer would be \(7\), because \(6\) is not included.
Solve it like an equation. Adding and dividing by a positive number leave the sign unchanged.
| \(5x - 2\) | \(>\) | \(13\) |
| \(5x\) | \(>\) | \(15\) |
| \(\therefore\ x\) | \(>\) | \(\boldsymbol{3}\) |
\(3\) itself is not a solution, so use an open circle.
Subtract \(5\) from both sides. Then divide by \(-2\): dividing by a negative number reverses the sign.
| \(5 - 2x\) | \(>\) | \(12\) |
| \(-2x\) | \(>\) | \(7\) |
| \(x\) | \(<\) | \(\dfrac{7}{-2}\) |
| \(\therefore\ x\) | \(<\) | \(\boldsymbol{-3.5}\) |
Check with \(x = -4\), which is in the solution: \(5 - 2(-4) = 13\), and \(13 > 12\) is true.
(i) \(\dfrac{2x - 1}{3} \ge x - 2\)
(ii) \(4 < \dfrac{x + 6}{2}\)
(i) Multiply both sides by \(3\). Then collect the \(x\) terms; the last step divides by \(-1\), so the sign reverses.
| \(2x - 1\) | \(\ge\) | \(3(x - 2)\) |
| \(2x - 1\) | \(\ge\) | \(3x - 6\) |
| \(-x\) | \(\ge\) | \(-5\) |
| \(\therefore\ x\) | \(\le\) | \(\boldsymbol{5}\) |
(ii) Multiply both sides by \(2\), subtract \(6\), then read the result from the \(x\) side.
| \(8\) | \(<\) | \(x + 6\) |
| \(2\) | \(<\) | \(x\) |
| \(\therefore\ x\) | \(>\) | \(\boldsymbol{2}\) |
\(2 < x\) and \(x > 2\) say the same thing. Turning the statement around turns the sign around.
The total spent must be less than or equal to $80.
| \(14.50 + 1.70n\) | \(\le\) | \(80\) |
| \(1.70n\) | \(\le\) | \(65.50\) |
| \(n\) | \(\le\) | \(38.53\ldots\) |
\(n\) must be a whole number, so round down. She can buy at most \(\boldsymbol{38}\) songs. Check: \(38\) songs cost $64.60, so $79.10 in total; \(39\) songs would bring the total to $80.80.
Common pitfalls
Frequently asked questions
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting any number, or multiplying or dividing by a positive number, leaves the sign as it is.
What is the difference between an open and a closed circle on a number line?
An open circle means the end number is not part of the solution, for \(<\) or \(>\). A closed circle means it is included, for \(\le\) or \(\ge\).
How many solutions does an inequality have?
Usually infinitely many. \(x > 3\) is true for every number bigger than \(3\), including decimals. In a word problem the context may limit it to whole numbers.
Why does the sign change when you divide by a negative?
Because multiplying by \(-1\) flips numbers through zero. \(2 < 5\), but \(-2 > -5\), so the order of the two sides is reversed.
How do you write at most and at least as inequalities?
At most means \(\le\). At least means \(\ge\). For example, spending at most $80 is \(\text{total} \le 80\).