What an inequality means
An inequality says one quantity is larger or smaller than another, rather than exactly equal. Where an equation has (usually) a single solution, an inequality has a whole range of solutions.
| Symbol | Means | Endpoint included? |
|---|---|---|
| \(<\) | less than | No (strict) |
| \(>\) | greater than | No (strict) |
| \(\le\) | less than or equal to | Yes |
| \(\ge\) | greater than or equal to | Yes |
Where this is assessed
Solving belongs to Criterion A, while writing the solution correctly on a number line and in set or interval form is pure Criterion C communication. Inequalities that model a real limit, such as a budget or a safe load, reach into Criterion D.
Solving multi-step inequalities
Treat the inequality sign just like an equals sign: whatever you do to one side, do to the other. Expand any brackets, gather the variable on one side and the numbers on the other, then divide to leave the variable on its own. As long as you only add, subtract, or multiply and divide by positive numbers, the sign stays exactly as it is.
Solve \(2(x - 3) + 4 > 3x - 1\).
Read it back the right way round
Ending with \(-1 > x\) is correct, but it is safer to rewrite it with \(x\) on the left: \(x < -1\). Keep the open end of the symbol pointing at the larger quantity and you will never misread which way it goes.
The flip: multiplying or dividing by a negative
Here is the one rule inequalities have that equations do not. Whenever you multiply or divide both sides by a negative number, you must reverse the inequality sign. To see why, start from the true statement \(3 > 1\). Multiply both sides by \(-1\) without flipping and you get \(-3 > -1\), which is false. Flipping the sign fixes it: \(-3 < -1\), which is true.
Solve \(-2x + 3 \ge 11\).
Only for multiply and divide
The flip applies to multiplying or dividing by a negative, never to simply adding or subtracting a negative. To dodge the whole issue, you can instead move the variable to the side where it stays positive: from \(-2x \ge 8\), add \(2x\) and subtract \(8\) to get \(-8 \ge 2x\), so \(x \le -4\) with no flip needed.
Representing the solution: number line, set and interval
Once solved, an inequality is usually shown in three ways. On a number line, an open circle marks a strict endpoint (\(<\) or \(>\), value not included) and a filled circle marks an included endpoint (\(\le\) or \(\ge\)); an arrow shows the direction of all the allowed values. Set notation describes the collection of solutions, and interval notation gives the two ends of the range, using a square bracket for an included end and a round bracket for an excluded one. Infinity always takes a round bracket.
| Inequality | Number line | Set notation | Interval notation |
|---|---|---|---|
| \(x < 3\) | Open circle at \(3\), arrow left | \(\{\,x : x < 3\,\}\) | \((-\infty,\ 3)\) |
| \(x \ge -2\) | Filled circle at \(-2\), arrow right | \(\{\,x : x \ge -2\,\}\) | \([-2,\ \infty)\) |
Solve \(\dfrac{x}{2} + 1 < 4\) and give the solution as a number line, in set form and in interval form.
Check yourself
Cover the answers and work each through, watching for a flip.
1. Solve \(3x - 7 > 8\). +
Add \(7\) to both sides: \(3x > 15\). Divide by \(3\) (positive, no flip): \(x > 5\).
2. Solve \(6 - 2x \le 14\). +
Subtract \(6\): \(-2x \le 8\). Divide by \(-2\) and flip the sign: \(x \ge -4\). Check \(x = -4\): \(6 - 2(-4) = 6 + 8 = 14\), and \(14 \le 14\) holds.
3. Express the solution \(x > 5\) on a number line and in interval notation. +
On the number line, draw an open circle at \(5\) (strict, so not included) with the arrow pointing right. In interval notation this is \((5,\ \infty)\), and in set notation \(\{\,x : x > 5\,\}\).
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