Key ideas
The four signs
> greater than < less than
≥ greater than or equal to (at least)
≤ less than or equal to (at most, no more than)
The small, pointed end points at the smaller number.
Number lines
Open circle ○: the number is not included (<, >).
Filled circle ●: the number is included (≤, ≥).
Solving
Solve an inequation just like an equation: do the same to both sides.
Keep the inequality sign where the = would be.
A negative x term
Add the x term to both sides so the x stays positive, then solve.
(If you divide by a negative number instead, the sign reverses.)
Open circle: 3 is not included.
Filled circle: −1 is included.
Typing answers: type x > 5, x <= -2 or x >= 3 (or use ≤ and ≥).
Notation and language
You must be at least 12 years old to go on a ride. Write this as an inequality, using a for age.(1 mark)
"At least" means 12 or more.a ≥ 12✓1
In plain English12 is allowed, and so is anything older. That is "greater than or equal to".
⚠ A common mistake: a > 12. That would mean a 12-year-old couldn't go on.
Inequalities on a number line
Write down the inequality shown on the number line.(1 mark)
Circle and direction.Filled circle at −2: −2 is includedArrow to the right: bigger numbersx ≥ −2✓1
In plain EnglishA filled circle means "or equal to". The arrow points to the numbers that work.
Considering solutions
List the whole numbers that satisfy −1 < x ≤ 3.(1 mark)
Check the ends.−1 is not included (<), 3 is included (≤)0, 1, 2, 3✓1
In plain Englishx has to be bigger than −1 and no more than 3.
Solving one-step inequations
Solve x + 7 > 12.(1 mark)
Take 7 from both sides.x > 12 − 7x > 5✓1
In plain EnglishExactly like solving x + 7 = 12, but the sign stays as it is.
Solving two-step inequations
Solve 3x − 4 ≤ 11.(2 marks)
Add 4 to both sides.3x ≤ 15✓1
In plain EnglishUndo the − 4 first.
Divide both sides by 3.x ≤ 153x ≤ 5✓2
In plain EnglishDivide both sides by 3. The sign stays the same because you are dividing by a positive number.
A negative x term
Solve 10 − 2x > 4.(2 marks)
Add 2x to both sides.10 > 4 + 2x6 > 2x✓1
In plain EnglishAdding 2x to both sides makes the x term positive. Then take 4 from both sides.
Divide, then read it from the x side.3 > xx < 3✓2
In plain English3 is bigger than x means x is less than 3. The sign still points at the smaller value.
The other way: −2x > −6. Dividing by −2 (a negative number) reverses the sign: x < 3. Same answer.
Multi-step inequations
Solve 3(x + 2) > x + 14.(3 marks)
Expand the bracket.3x + 6 > x + 14✓1
In plain EnglishMultiply both terms in the bracket by 3.
Collect x terms and numbers.3x − x > 14 − 62x > 8✓2
In plain EnglishTake x from both sides, and 6 from both sides.
Divide.x > 4✓3
In plain EnglishDivide both sides by 2. The sign stays the same because 2 is positive.
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Your turn
Type answers like x > 5 or x <= -2 (use < > <= >= or ≤ ≥). Stuck? Tap Need a hint?