🎯 Learning Objectives
- Solve separate linear inequalities.
- Find the overlap of their solution sets.
- Write combined answers clearly.
Introduction
This lesson develops Simultaneous Inequalities from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
🤔 Start with an Idea
A student may need to be at least 12 years old but younger than 18. Both conditions must be true. Simultaneous inequalities describe this shared range.
📖 Key Notes
Solve each inequality, then find the intersection of the answer ranges.
x ≥ 2 and x < 7 becomes 2 ≤ x < 7
📏 Overlap Explorer
The filled point includes 2. The open point excludes 7.
🧠 Let's Think Together
Solve: 3x − 1 ≥ 5 and 2x + 3 < 13.
- 3x ≥ 6, so x ≥ 2.
- 2x < 10, so x < 5.
- The overlap is from 2 up to, but not including, 5.
2 ≤ x < 5
✍️ Worked Examples
−1 < 2x + 3 ≤ 9
−4 < 2x ≤ 6
−2 < x ≤ 3
−4 < 2x ≤ 6
−2 < x ≤ 3
🎯 Practice Questions
Easy
Write x > 1 and x ≤ 6 as one compound inequality.
Show answer
1 < x ≤ 6
Challenge
Solve −5 ≤ 3x + 1 < 10.
Show answer
−2 ≤ x < 3
⚠️ Common Mistakes
- Taking the union when the question requires both conditions.
- Forgetting to reverse an inequality when dividing by a negative.
- Using a filled point for a strict inequality.
📝 Exam Focus
Use a number line to check the overlap. If the regions never overlap, state that there is no solution.
Summary
- Solve every condition.
- Reverse the sign after multiplying or dividing by a negative.
- Keep only the overlap.