Mathematics from zero
Inequalities
A sign at a ride says “you must be at least 12 years old”. That is not one age — a 12-year-old qualifies, so does a 13-year-old, so does a 40-year-old. The rule describes a whole range of ages at once. Mathematics writes such a rule as an inequality.
After this lesson you can say what an inequality is, read the four comparison signs, understand that an inequality’s answer is a range, and solve a simple inequality the same way you solve an equation.
An inequality compares two sides instead of equating them. An equation uses =.
An inequality uses a comparison sign: > (greater than), < (less than), ≥ (at
least — greater than or equal to), or ≤ (at most — less than or equal to). x > 4
says “x is some number bigger than 4”.
An inequality’s answer is a range of numbers, not a single one. Solving x + 3 = 7
gave exactly one value, x = 4. But x > 4 is true for 5, for 6, for 7, for
every number past 4. The solution of an inequality is the whole set of values that
make it true — a range, not a point.
The number line above shows the range. The inequality x > 4 is marked: every
value to the right of 4 is highlighted, because every one of them satisfies “bigger
than 4”. The single number 4 itself is not highlighted — > means strictly bigger.
Had the sign been ≥, the 4 would join the range too.
Solve an inequality just like an equation. The balance rule still holds: whatever
you add to, subtract from, multiply, or divide one side by — using positive numbers —
do to the other side, and the inequality stays true. So solve x + 3 > 7 by
subtracting 3 from both sides, exactly as for an equation, reaching x > 4.
Solve 2x + 1 > 9.
Treat it like the equation 2x + 1 = 9, but keep the > sign throughout.
Undo the + 1: subtract 1 from both sides. The inequality becomes 2x > 8.
Undo the × 2: divide both sides by 2. The inequality becomes x > 4.
The solution is x > 4 — every number greater than 4. Test one: x = 5 gives
2 × 5 + 1 = 11, and 11 > 9 is true. Test a number not in the range: x = 4 gives
9, and 9 > 9 is false — correctly excluded.
Why this works
Why is the answer a range and not a single value? Because the question itself is looser. An equation pins the variable to one exact spot — “where is the balance point?” An inequality only asks “which side of the boundary?” Many numbers lie on the same side, so many numbers answer it. The looser question naturally has a wider answer.
Common mistake
A common mistake is reporting a single number as the answer — writing the solution of
x > 4 as “5”. 5 is one solution, but so is 6, and 7, and 100. The full answer is
the range x > 4. Naming one value from the range and stopping leaves out almost all
of the answer.
Solve x + 2 > 10. The solution is x > a boundary number — type that boundary.
Solve x − 1 < 5. The solution is x < a boundary — type that boundary.
In the inequality x ≥ 7, what is the smallest value x is allowed to be? Type it.
Solve 3x > 12. The solution is x > a boundary — type that boundary.
Is 9 a solution of x > 5? Type 1 for yes, 0 for no.
You solve an inequality and reach x > 4. What is the solution?
An inequality compares two sides with >, <, ≥, or ≤ instead of =. Its answer
is not a single number but a whole range — every value on the correct side of the
boundary. You solve an inequality the same way as an equation, applying the balance
rule with positive numbers, until the variable stands alone. The result is a range,
and the full range is the answer.