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Hong KongInequalities and their solution sets
Solve an inequality exactly as you would an equation, with one addition: multiplying or dividing both sides by a negative flips the direction of the sign. The answer is a range rather than a value, and items often show it on a number line.
The rule pairs inequalities with equations three separate times, in skills 2, 3 and 4. They are not a footnote to the algebra competency.
The one extra rule
Multiply or divide both sides by a negative and the sign reverses. Everything else works exactly as it does for an equation, which is worth saying plainly because the reputation of inequalities suggests otherwise.
Take negative 2x plus 3 greater than 11. Subtract 3 to get negative 2x greater than 8. Divide by negative 2 and the sign flips: x is less than negative 4.
Test it. Try x equals negative 5, which satisfies the answer. Negative 2 times negative 5 is 10, plus 3 is 13, which is greater than 11. The answer holds, and the ten seconds that took would have caught a missing flip.
The four signs and their words
| Sign | Reads as | In a word problem |
|---|---|---|
| Less than | strictly below | under, fewer than, below |
| Less than or equal to | at most | no more than, a maximum of, up to |
| Greater than | strictly above | over, more than, exceeds |
| Greater than or equal to | at least | no fewer than, a minimum of |
The middle two rows decide items. At most includes the endpoint. Under does not. A budget of at most 100 dollars allows spending exactly 100, and a question that hinges on that difference will offer you both readings as options.
Reading a number line
- An open circle excludes the endpoint, which means a strict inequality
- A closed circle includes it, which means an or-equal-to inequality
- The shading shows the direction, and the arrow shows it continues
A question can give you the inequality and ask for the line, or give you the line and ask for the inequality. Both are settled by those three bullets, and the circle is the part that separates two otherwise identical options.
A field trip requires at least 3 chaperones, and one chaperone is needed for every 8 students. If s is the number of students, which inequality models the requirement?
- s divided by 8 is less than 3
- s divided by 8 is at least 3
- s divided by 8 is at most 3
- s times 8 is at least 3
Compound inequalities
Two conditions at once, joined by and or or. An and gives a band between two values; an or gives everything outside a band.
Written as a chain, such as 2 less than x less than 10, it always means and. Do the same operation to all three parts and the chain stays true. That makes solving one no harder than solving a single inequality, which surprises people.
Where inequalities show up outside algebra
In word problems about budgets, capacity, minimum requirements and thresholds, which is most of what a real-world mathematics item is about. The rule names inequalities alongside equations in the modeling skill precisely because a constraint is usually an inequality rather than an equation.
Our view: the flip rule gets all the attention and the endpoint language causes more lost marks. At most and fewer than are not the same, and no amount of algebraic care will save you if you translated the sentence wrongly at the start.
Common questions
When does an inequality sign flip?
When you multiply or divide both sides by a negative number. Adding, subtracting, and multiplying or dividing by a positive all leave the direction alone. Substituting a value from your answer back into the original will catch a missed flip.
What does at most mean in an inequality?
Less than or equal to, so the endpoint is included. A budget of at most 100 dollars permits spending exactly 100. Compare that with under 100, which excludes it, and the difference decides which of two otherwise identical options is right.
What do open and closed circles mean on a number line?
An open circle excludes the endpoint and corresponds to a strict inequality. A closed circle includes it and corresponds to an or-equal-to inequality. The shading gives the direction and an arrow shows the range continues.
Are inequalities really on the FTCE?
Yes, and prominently. The rule names them alongside equations in three of the seven algebra skills: modeling a real-world situation, solving graphically or algebraically, and determining and solving in real-world problems.