Revision notes
Read and write inequalities with <, >, ≤ and ≥, solve them like equations, and show the answers on a number line.
At the amusement park in Abuja, Obi is 11, so he goes in free — but his sister Zainab, who is 12, must pay. "Under 12" is not one number; it is a whole range of ages. Maths writes it as an inequality: age < 12.
x < 5 means x is less than 5. x > 5 means x is greater than 5.
x ≤ 5 means x is less than or equal to 5, so 5 itself is included. x ≥ 5 means greater than or equal to 5.
On a number line, smaller numbers are on the left. So −3 < 2, because −3 is to the left of 2.
🔑 Key idea: The small, pointed end of < or > faces the smaller number.
Draw an open (empty) circle when the number is NOT included (< or >) and a solid dot when it IS included (≤ or ≥).
Then draw an arrow left for 'less than' or right for 'greater than'. x ≥ −2 is a solid dot at −2 with an arrow to the right.
The integers that satisfy −2 ≤ x < 3 are −2, −1, 0, 1 and 2 — the 3 is left out because of the <.
🔑 Key idea: Open circle = not included. Solid dot = included.
Solve it just like an equation, keeping the sign instead of =. For 2x + 3 < 11: subtract 3 from both sides to get 2x < 8, then divide by 2 to get x < 4.
Adding, subtracting, or multiplying and dividing by a positive number never changes the sign.
Careful! If you multiply or divide by a negative number, you must reverse the sign. For −2x < 6, dividing by −2 gives x > −3. (Check: x = 0 works, because −2 × 0 = 0 < 6.)
🔑 Key idea: Treat it like an equation — but divide by a negative and the sign flips.
Solve the inequality x + 12 < 19.
Step 1: Subtract 12 from both sides
x + 12 − 12 < 19 − 12
x < 7
Step 2: Say it in words
x is less than 7.
Answer: x < 7
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