Double inequalities
Objective. Represent values satisfying two linked bounds.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A double inequality links a variable between two bounds in a single statement, such as 2 < x ≤ 7, meaning x is greater than 2 and less than or equal to 7 simultaneously. To solve a double inequality involving an expression, apply the same operation to all three parts at once, keeping the inequality signs consistent throughout. For example, to solve 3 < 2x + 1 ≤ 9, subtract 1 from all three parts to get 2 < 2x ≤ 8, then divide by 2 to get 1 < x ≤ 4. As with single inequalities, multiplying or dividing by a negative number reverses the direction of both inequality signs.
Worked example
Solve the double inequality 5 ≤ 3x − 1 < 14 for x.
- 1Add 1 to all three parts: 6 ≤ 3x < 15.
- 2Divide all three parts by 3.
- 36 ÷ 3 = 2 and 15 ÷ 3 = 5.
- 4This gives 2 ≤ x < 5.
Answer: 2 ≤ x < 5
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils sometimes apply an operation to only one side of the double inequality, such as subtracting 1 from the middle and right parts but forgetting the left, which breaks the balance. Always apply the same operation to all three parts together.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.