Linear Functions, Equations and Inequalities
Senior 1 · Mathematics
Linear Functions, Equations and Inequalities
Lesson 5 of 5
25 min
Let's begin
Ishimwe has 2,000 Frw. He must keep 200 Frw for airtime, and suppose each daily data bundle costs 300 Frw. How many bundles can he buy? The answer is not one number: he can buy 0, 1, 2… up to some largest number. Questions with "at most", "at least", "no more than" or "more than" lead to inequalities, and their answers are whole ranges of numbers.
Definition
Inequality: a statement that one quantity is less than, greater than, at most or at least another. A linear inequality in one unknown is like a linear equation with the sign replaced by , , or . Example: .
Its solution set is the set of all values that make it true.
Definition
Interval: all the real numbers between two endpoints. A round bracket leaves an endpoint out and a square bracket puts it in. Example: means every number greater than and less than or equal to 3, that is .
| Symbol | Read as | Circle on the number line | Interval |
|---|---|---|---|
| is greater than | open, arrow to the right | ||
| is at least | closed, arrow to the right | ||
| is less than | open, arrow to the left | ||
| is at most | closed, arrow to the left | ||
| is between and , including | open at , closed at |
An open circle (and a round bracket) means the endpoint is not included; a closed circle (and a square bracket) means it is. The symbol (infinity) is not a number, so it always takes a round bracket.
Solve a linear inequality exactly like an equation, with one exception:
Important
Why the sign reverses. Start from . Multiply both numbers by to get and . On the number line, is to the right of , so . Multiplying by a negative number reflects every point in 0, which swaps left and right, and so the order is reversed.
Worked example
Solve . Give the answer as an interval and on a number line
Interval: . The divisor 3 is positive, so the sign is unchanged.
Worked example
Solve
Interval: . Test a value: gives ✓, while gives ✗, as the answer predicts.
Worked example
Solve in two ways. (Brackets.)
Remove the bracket: .
Way 1, unknowns on the left: . Divide by and reverse: .
Way 2, unknowns on the right, so nothing is divided by a negative: , so , so .
Both ways give . This is a good check on the reversal rule: without it, Way 1 would give the wrong answer .
Worked example
Solve . (Fractions.)
The LCM of 2 and 3 is 6, a positive number, so the sign stays:
Interval: .
Worked example
Solve and list the integers in the solution set
Do the same thing to all three parts:
Interval: . The integers are ; is excluded by the open circle.
Worked example
Back to Ishimwe: how many 300 Frw bundles can he buy if he keeps 200 Frw of his 2,000 Frw?
Let be the number of bundles. The total he spends must be at most 2,000 Frw:
counts bundles, so it must be a whole number: . He can buy at most 6 bundles.
Warning
Reverse the sign only when you multiply or divide by a negative number. In you divide by , so . But in you divide by , so : a negative number on the other side does not reverse anything.
Warning
Read the context. If the unknown counts people, sacks or bundles, the answer must be a whole number. "At most 6.8 bundles" means at most 6, since 7 would cost too much.
Tip
Test your answer: pick one number inside your solution set and one outside it, and put each into the original inequality. The inside number should make it true and the outside number false.
Math Antics (mathantics): "Algebra Basics: Inequalities In Algebra". Watch for the part that explains why the inequality sign flips when you multiply or divide by a negative number.
Practice
Solve each inequality. Show the answer on a number line where asked.
Further reading
Sources and credits
The data-bundle and lift problems, the worked examples, diagrams, practice items and quiz questions were written by Ganzaa. The bundle price is a model for practice, not a network's tariff.
