Solving Inequalities — Print
Solve them exactly like equations — with one extra rule that catches almost everyone.
What the practice questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- Inequality — A statement that one expression is less than or greater than another.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Range — The set of outputs that actually occur.
- Number line — A line with numbers increasing to the right.
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.
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.
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.
- Subtract 3: -4x ≥ 16 Subtracting does not flip anything.
- Divide by -4 and FLIP: x ≤ -4 Dividing by a negative reverses the inequality.
- Check with x = -5: -4(-5) + 3 = 23 ≥ 19 ✓ A value in the range must work.
- 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?
- Let t = T-shirts. Write 28 + 14t ≤ 85 She cannot spend more than she has.
- 14t ≤ 57 Subtract 28 from both sides.
- t ≤ 4.07… Divide by 14 — positive, so no flip.
- Round down to 4. She cannot buy part of a shirt.
At most 4 T-shirts.
Compound inequality
Solve -7 ≤ 2x - 1 < 9.
- Add 1 to all three parts: -6 ≤ 2x < 10 Whatever you do, do it everywhere.
- Divide all three by 2: -3 ≤ x < 5 Positive divisor, so nothing flips.
- 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.
Problem 1
Solve x + 5 > 12. Enter your answer like x>7.
Worked answer
x>7
x > 7
Problem 2
Solve 3x ≤ 21. Enter your answer like x<=7.
Worked answer
x<=7
x ≤ 7
Problem 3
Solve x - 4 ≥ -1. Enter your answer like x>=3.
Worked answer
x>=3
x ≥ 3
Problem 4
On a number line, which dot do you use for x > 5?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
An open (hollow) dot
x > 5 does not include 5 itself, so the dot is open.
Problem 5
Solve x2 > 6. Enter your answer like x>12.
Worked answer
x>12
x > 12
Problem 6
Solve 3x + 4 < 19. Enter your answer like x<5.
Worked answer
x<5
3x < 15
x < 5
Problem 7
Solve -4x + 3 ≥ 19. Enter your answer like x<=-4.
Worked answer
x<=-4
-4x ≥ 16
Divide by -4 and flip: x ≤ -4
Check x = -5: 20 + 3 = 23 ≥ 19 ✓
Problem 8
Solve 5 - 2x > 11. Enter your answer like x<-3.
Worked answer
x<-3
-2x > 6
Divide by -2 and flip: x < -3
Problem 9
Solve 2(x - 3) ≤ 8. Enter your answer like x<=7.
Worked answer
x<=7
x - 3 ≤ 4
x ≤ 7
Problem 10
Ava has $85. She buys a $28 jacket, then T-shirts at $14 each. At most how many T-shirts?
Worked answer
4
14t ≤ 57
t ≤ 4.07…
She can buy 4 shirts.
Problem 11
Which dot do you use for x ≥ 3?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
A closed (filled) dot
3 itself is a solution, so the dot is closed.
Problem 12
Solve 7x - 2 > 4x + 13. Enter your answer like x>5.
Worked answer
x>5
3x - 2 > 13
3x > 15
x > 5
Problem 13
How many integers satisfy -7 ≤ 2x - 1 < 9?
Worked answer
8
Add 1 everywhere: -6 ≤ 2x < 10
Divide by 2: -3 ≤ x < 5
Integers: -3, -2, -1, 0, 1, 2, 3, 4
That is 8 values — note 5 is excluded.
Problem 14
Solve -3(x - 2) < 12. Enter your answer like x>-2.
Worked answer
x>-2
-3x + 6 < 12
-3x < 6
Divide by -3 and flip: x > -2
Problem 15
Priya scored 76, 84 and 79 on three tests. What is the lowest fourth score that gives an average of at least 80?
Worked answer
81
Needed total: 80 × 4 = 320
Current total: 76 + 84 + 79 = 239
320 - 239 = 81
Problem 16
Solve x-5 ≥ 3. Enter your answer like x<=-15.
Worked answer
x<=-15
Multiply both sides by -5 and flip: x ≤ -15
Check x = -20: -20-5 = 4 ≥ 3 ✓