Solving Inequalities

Solve them exactly like equations — with one extra rule that catches almost everyone.

A lift is rated for 600 kg. Six people get in, averaging 72 kg each.

That is 432 kg, so the lift is fine. But notice what the sign did not say.

a lift that carries at most 600 kg 432 kg limit 600 kg six people averaging 72 kg 168 kg spare the limit is not a weight — it is a line you must not cross

It did not say the lift weighs 600 kg. It said 600 is a line you must not cross.

Any load from 0 up to 600 is allowed — a whole range, not one number. In symbols, w ≤ 600.

So the answer to "how much more can it take?" is also a range: anything up to 168 kg more.

That is the difference from an equation. An equation pins down a point; an inequality fences off a region. And one rule is new: multiply or divide by a negative and the fence flips sides.

Number line showing x is less than or equal to negative four -8-7-6-5-4-3-2-101234 x ≤ -4  ·  closed dot includes -4; shading runs left forever -4 is a solution an open circle would mean x < -4, leaving -4 out

Same moves as equations

Add, subtract, multiply, divide — all the familiar steps still work, and you still do them to both sides.

3x + 4 < 19 ⇒ 3x < 15 ⇒ x < 5. Identical to solving 3x + 4 = 19, except the answer is a whole range.

The one extra rule

When you multiply or divide both sides by a negative number, the inequality sign flips.

Here is why, with real numbers. Start with a true statement: 4 < 10. Multiply both sides by -1: you get -4 and -10. But -4 is greater than -10 — on the number line, -4 sits to the right. So the correct statement is -4 > -10. The sign had to turn around.

Multiplying by a negative reflects the number line, and reflection reverses order.

Flip only for negative × or ÷

Adding or subtracting a negative does not flip the sign. Only multiplying or dividing by a negative does. x - 5 > 2 stays >; -2x > 8 becomes x < -4.

Graphing on a number line

Draw the boundary point. Use an open circle for < or >, a closed circle for ≤ or ≥.

Shade the side that contains the solutions. Test a point if you are unsure: for x < 5, try x = 0 — it works, so shade the side containing 0.

Compound inequalities

3 < x ≤ 9 means x is both greater than 3 and at most 9. Graph it as a segment with an open left end and a closed right end.

Solve them by doing the same step to all three parts: 2 < x + 5 ≤ 11 becomes -3 < x ≤ 6 by subtracting 5 everywhere.

Inequalities in stories

"At least", "minimum", "no fewer than" → ≥.
"At most", "maximum", "no more than", "up to" → ≤.

Then apply common sense to the answer. If x ≤ 7.3 counts tickets, the real answer is 7.

Worked examples

The flip in action

Solve -4x + 3 ≥ 19 and graph the solution.

  1. Subtract 3: -4x ≥ 16 Subtracting does not flip anything.
  2. Divide by -4 and FLIP: x ≤ -4 Dividing by a negative reverses the inequality.
  3. Check with x = -5: -4(-5) + 3 = 23 ≥ 19 ✓ A value in the range must work.
  4. Graph: closed dot at -4, shade left. ≤ includes the endpoint.

x ≤ -4

A budget

Ava has $85. She buys a $28 jacket and wants T-shirts at $14 each. How many can she buy?

  1. Let t = T-shirts. Write 28 + 14t ≤ 85 She cannot spend more than she has.
  2. 14t ≤ 57 Subtract 28 from both sides.
  3. t ≤ 4.07… Divide by 14 — positive, so no flip.
  4. Round down to 4. She cannot buy part of a shirt.

At most 4 T-shirts.

Compound inequality

Solve -7 ≤ 2x - 1 < 9.

  1. Add 1 to all three parts: -6 ≤ 2x < 10 Whatever you do, do it everywhere.
  2. Divide all three by 2: -3 ≤ x < 5 Positive divisor, so nothing flips.
  3. Graph: closed at -3, open at 5, shade between. The symbols tell you which ends are included.

-3 ≤ x < 5

Words to know

Inequality symbols
< less than, > greater than, ≤ at most, ≥ at least.
Solution set
All the values that make the inequality true — usually infinitely many.
Open / closed dot
Open excludes the endpoint (<, >); closed includes it (≤, ≥).
Compound inequality
Two conditions at once, like 3 < x ≤ 9.

Mistakes to watch for

Forgetting to flip when dividing by a negative — the single most common error in this unit. -2x > 8 becomes x < -4, because negating reverses the order of the number line: 3 < 5 is true, -3 < -5 is false.

Flipping when you only added or subtracted a negative. Adding never changes the direction, however many minus signs appear. Only multiplying or dividing by a negative does.

Using an open circle where the endpoint counts. ≤ includes the boundary and gets a filled circle; < excludes it and gets a hollow one. That tiny hole is the whole difference between the two.

Rounding the wrong way in a word problem. r ≤ 9.5 rides means 9 rides, not 10 — round towards what the situation allows, not by the usual rule.

Lift weight limits are set with a large safety margin — the cables in a modern passenger lift are typically rated for many times the stated load.

The reason anyone trusts a lift at all is a piece of theatre: in 1854 Elisha Otis stood on a raised platform at the New York World's Fair and had the rope cut.

His safety brake caught it. He sold lifts afterwards, and the word "elevator" entered ordinary use.

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