Linear Inequalities
Solve it like an equation, with one extra rule: multiplying or dividing by a negative reverses the sign.
The sign at the ride says: you must be under 1.4 metres to ride.
That is not one height. It lets in a whole range of people — and shuts out another range.
On a number line the allowed heights are a stretch, not a point. That is the first thing an inequality does differently from an equation.
Look at the circle at 1.4. It is hollow, because 1.4 exactly is not allowed — "under" excludes it.
Change the sign to "1.4 or under" and that circle fills in. One tiny hole is the whole difference between < and ≤.
Solving works exactly like an equation, with one extra rule. Multiply or divide by a negative and the direction reverses — because 3 < 5 is true, but -3 < -5 is false.
The one extra rule
Adding or subtracting never affects the direction. Multiplying or dividing by a positive never affects it either.
Multiplying or dividing by a negative reverses it. -2x > 6 becomes x < -3.
Why: 3 < 5 is true, but -3 < -5 is false. Negating flips the order of the number line.
Flip only for a negative multiply or divide
Subtracting a negative does not flip anything. Only multiplying or dividing both sides by a negative number does.
Graphing on a number line
An open circle for < or >: the endpoint itself is not included.
A filled circle for ≤ or ≥: the endpoint is included.
Shade in the direction of the solutions, and test one value from the shaded part to be sure.
Test a point
After solving, pick any number in your shaded region and put it into the original inequality. If it works, the direction is right.
Compound inequalities
"And" means both must hold, so you take the overlap. -2 < x ≤ 5 is the stretch between them.
"Or" means either will do, so you take everything in both pieces — often two separate arrows.
Solve a three-part inequality by doing the same thing to all three parts at once.
"And" is an overlap, "or" is a union
x > 3 and x > 7 is just x > 7 — only the overlap survives. x > 3 or x > 7 is x > 3 — the union.
Inequalities in words
"At least" is ≥. "At most" is ≤. "More than" is >. "No more than" is ≤.
"A budget of $50" means the total must be ≤ 50.
Then check the context: if the variable counts people or tickets, the answer must be a whole number, so round down for a maximum.
Round the right way
If n ≤ 7.4 tickets, the answer is 7, not 8 — even though 7.4 rounds to 7 only by luck. Always round towards what the context allows.
Worked examples
A negative coefficient
Solve -3x + 5 ≥ 17 and graph it.
- Subtract 5: -3x ≥ 12 Subtracting never flips the sign.
- Divide by -3 and flip: x ≤ -4 Dividing by a negative reverses the direction.
- Graph: filled circle at -4, shade left. ≤ includes the endpoint.
- Test x = -10: -3(-10) + 5 = 35 ≥ 17 ✓ Confirms the direction.
x ≤ -4
Compound
Solve -7 < 2x - 1 ≤ 9.
- Add 1 to all three parts: -6 < 2x ≤ 10 Whatever you do, do it to all three.
- Divide all three by 2: -3 < x ≤ 5 Dividing by a positive keeps both directions.
- Graph: open circle at -3, filled at 5, shade between. Different endpoints, different circles.
-3 < x ≤ 5
In context
A ride costs $4 and entry is $12. With $50, how many rides can you take?
- Let r be the number of rides: 12 + 4r ≤ 50 "With $50" means the total is at most 50.
- Subtract 12: 4r ≤ 38 Undo the entry fee.
- Divide by 4: r ≤ 9.5 Positive divisor, so no flip.
- Rides are whole, so at most 9. Check: 12 + 36 = $48 ≤ $50, but 10 rides would be $52 ✓
9 rides
Words to know
- Inequality
- A statement using <, >, ≤ or ≥.
- Solution set
- Every value that satisfies the inequality — usually an interval, not a point.
- Compound inequality
- Two conditions joined by "and" or "or".
- Interval notation
- A compact way of writing a solution set, like [3, ∞).
Mistakes to watch for
Flipping the sign when merely subtracting a negative. Only multiplying or dividing by a negative reverses the direction. Adding and subtracting never do, however many minus signs are on the page.
Forgetting to flip after dividing by a negative. -2x > 8 becomes x < -4. Test it: x = -5 gives -2(-5) = 10 > 8, which works — and x = 0 does not.
Using an open circle where the endpoint counts. ≤ and ≥ include the boundary, so the circle is filled. It is the difference between being allowed on the ride and not.
Leaving a fractional answer where the context demands whole things. r ≤ 9.5 rides means 9 rides. Round towards what the situation allows, not by the usual rounding rule.
The signs < and > first appear in a book by Thomas Harriot, published in 1631 — ten years after he died.
Harriot was an astronomer too. He drew the Moon through a telescope several months before Galileo, but published almost nothing in his lifetime.
The symbols ≤ and ≥ came much later, from the French mathematician Pierre Bouguer in 1734.