Linear inequalities and number-line solutions · Higher
| English | 中文 | Pinyin |
|---|---|---|
| solution set/səˈluːʃn set/ | 解集 | jiě jí |
Which purchases fit within the budget?
- A student can spend at most 17 yuan on a fixed 5-yuan charge plus 3 yuan per item. Is one equality enough to describe every allowed purchase?
- This lesson studies solution set 解集: All inputs that satisfy the stated condition.
Choose the mathematical structure
- Solve a linear inequality using the same balance operations as an equation. Multiplying or dividing by a negative reverses its direction. Use a closed endpoint for ≤ or ≥ and an open endpoint for < or >. Intersect restrictions to find their common permitted inputs.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines solution set?
All inputs that satisfy the stated condition.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
3x+5≤17 gives x≤4. For nonnegative whole items, the permitted values are 0,1,2,3,4. Also -2x<6 gives x>-3 after division by -2. The combined restriction -3<x≤4 has an open circle at -3 and a closed circle at 4. A value x=5 fails the original budget because 3×5+5=20.
Linear inequalities and number-line solutions
Solve a linear inequality using the same balance operations as an equation
Compare the model with the worked case and explain one change.
Find the largest whole x satisfying 3x+5≤17.
Subtract 5 then divide by 3: x≤4.
Test a tempting shortcut
- A reversed sign is needed only when multiplying or dividing by a negative, not when adding a negative. Include the physical domain: negative or fractional item counts may be meaningless.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Dividing an inequality by a negative number leaves its direction unchanged. This claim is false. Explain which definition or assumption it violates.
Find the boundary value after solving -2x<6.
Divide by -2 and reverse: x>-3.
Dividing an inequality by a negative number leaves its direction unchanged.
A reversed sign is needed only when multiplying or dividing by a negative, not when adding a negative. Include the physical domain: negative or fractional item counts may be meaningless.
Interpret a new situation
- AQA A22 Foundation requires one-variable linear inequalities and number lines. Higher quadratic and two-variable regions are in a separate lesson; avoid replacing the inequality with a single boundary value.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many nonnegative whole-item counts satisfy the budget?
List 0,1,2,3,4: five counts.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
All inputs that satisfy the stated condition. Choose the relationship, show the method, check its assumptions and interpret the result.