Inequalities

A lot of everyday life isn't an exact equation: your shopping may cost at most 20 euros, you must be at least 12 years old for the movie, the parcel has to weigh less than 5 kilos. Mathematics describes conditions like these with inequalities, and you'll see: you can work with them almost exactly like with equations.

Four signs for four conditions

Instead of the equals sign you use comparison signs: x < 5 means x is less than 5. x > 5 means x is greater than 5. x ≤ 5 means x is at most 5 (the 5 itself counts too). x ≥ 5 means x is at least 5. The tip of the sign always points to the smaller side.

Solution sets instead of one solution

An equation like x + 2 = 7 has exactly one solution: x = 5. The inequality x + 2 < 7, on the other hand, has infinitely many solutions: every number smaller than 5 works, so 4, but also 0, minus 3 or 4.9. That's why you write the solution as a condition: x < 5. All numbers that meet this condition together form the solution set.

Solve like an equation, with one exception

You may add or subtract the same number on both sides and multiply or divide by the same positive number, just like with equations. Only one thing is different: if you multiply or divide by a negative number, the comparison sign flips. An example: from −2x < 10, dividing by −2 gives the solution x > −5. Remember this exception, it's the only pitfall.

From words to an inequality

Most of the time the inequality is still hidden in a sentence. Example: a taxi ride costs a 4 euro base fare plus 2 euros per kilometre, and you have 20 euros in your pocket. Call the kilometres travelled x, then the condition reads 4 + 2x ≤ 20. Subtract 4 on both sides, which gives 2x ≤ 16, divide by 2, and you get x ≤ 8. So eight kilometres are within budget.