Practical applications and areas where solutions overlap

Translating a problem into an inequality

Inequalities are often used to model real-life situations involving constraints such as ‘at least’, ‘at most’ or ‘not exceeding’.

Example: A bike hire package costs a fixed fee of 5 euros plus 2 euros per hour. How many hours can you hire the bike for a maximum budget of 23 euros?

Let x be the number of hours. The total cost is 5 + 2x. We want 5 + 2x ≤ 23.

• 2x ≤ 23 - 5
• 2x ≤ 18
• x ≤ 9

You can hire the bike for a maximum of 9 hours.

Intersection of two conditions

Sometimes a problem involves two inequalities at the same time. Example: x must satisfy x > 2 AND x ≤ 8. The solution is then the intersection of the two intervals: ]2; 8].

On a number line, we plot each condition and then keep only the part common to both.

Example with fractions

Let’s solve x/3 - 1 >= 2:

• x/3 >= 2 + 1
• x/3 >= 3
• x >= 9 (we multiply by 3, which is positive; the direction does not change)

Common pitfall

In a real-world problem, always check whether the variable has any additional constraints (for example, x must be a positive integer, or represent a number of hours, so x ≥ 0). Don’t forget to relate the mathematical answer to the context of the problem.