Linear inequalities in one unknown

Skills and methods in this topic, easy to hard

1. 1Inequalities and the number line> < hollow, ≥ ≤ filled; greater runs right, less runs left

   Methodx ≥ −2: a filled circle at −2, running right

   Another way: Put a number inx ≥ −2: −2 ≥ −2 is true, so −2 is a solution

2. 2Properties of inequalitiesTimes or divided by a negative: turn the sign round

   Method4 > 1, times −2: −8 < −2

   Another way: Move x to where it is positive−2x > 6 ⇒ −6 > 2x ⇒ x < −3

3. 3Solving linear inequalitiesSolve as an equation, turning the sign when dividing by a negative; check one solution

   Method5 − 2x ≤ 11: −2x ≤ 6, x ≥ −3

   Another way: x to the positive side−6 ≤ 2x, x ≥ −3

4. 4Compound inequalities: and, or"And": where they overlap. "Or": both parts together

   Methodx > −1 and x ≤ 4: −1 < x ≤ 4

   Another way: A number from each stretchx = 0: 0 > −1 and 0 ≤ 4, both true

5. 5Word problems with inequalitiesAt least ≥, at most ≤; at most rounds down, at least rounds up

   Method6x ≤ 50: x ≤ 8.33…, at most 8 pens

   Another way: Find the boundary, then check6 × 8 = 48 ≤ 50, 6 × 9 = 54 > 50

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Inequalities and the number line →

   Which inequality does the number line show?

   • A. x < −4
   • B. x ≥ −4
   • C. x ≤ −4
   • D. x > −4

   Choose the right answer

   ▶Show the answer and working

   Answer: D. x > −4

   Look at two things: is the circle hollow (−4 left out) or filled (−4 in), and does the line run right (greater) or left (less)?

2. Example 2Solving linear inequalities →

   Solve 6x + 1 ≥ 31: x ≥

   ▶Show the answer and working

   Answer: 5

   Take 1 from both sides: 6x ≥ 31 − (1) = 30; divide both sides by the positive 6, the sign staying: x ≥ 5.

3. Example 3Word problems with inequalities →

   With $35, at most how many pens at $4 each can be bought?

   ▶Show the answer and working

   Answer: 8 pens

   With x pens: 4x ≤ 35, x ≤ 354. A count is whole, so round down: at most 8.