Quadratic inequalities and two-variable regions · Higher
| English | 中文 | Pinyin |
|---|---|---|
| boundary line/ˈbaʊndəri laɪn/ | 边界线 | biān jiè xiàn |
Where do all the restrictions overlap?
- A design must lie above one line and below another. How can a graph show every feasible pair at once?
- This lesson studies boundary line 边界线: The equality line separating values that satisfy an inequality from those that do not.
Choose the mathematical structure
- For a quadratic inequality, locate roots and test the sign in each interval. For a two-variable inequality, draw its equality boundary; use a dashed line if equality is excluded and a solid line if included. Test a point on each side and intersect the allowed regions. Write the domain or set notation clearly.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines boundary line?
The equality line separating values that satisfy an inequality from those that do not.
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.
(x-3)(x-7)≤0 holds on 3≤x≤7, since the factors have opposite signs between the roots and the endpoints give zero. For y≥x+1 and y<5, shade on/above the solid line y=x+1 and below the dashed line y=5. Their meeting input is x=4, but (4,5) is excluded by y<5. The point (1,3) satisfies both inequalities.
Quadratic inequalities and two-variable regions
For a quadratic inequality, locate roots and test the sign in each interval
Compare the model with the worked case and explain one change.
Find the lower endpoint of (x-3)(x-7)≤0.
The smaller zero is 3, and it is included.
Test a tempting shortcut
- For an upward-opening quadratic, positive values lie outside the roots, not between them. A boundary intersection is not necessarily included. Use a test point not on the boundary itself.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A dashed inequality boundary includes every point on its line. This claim is false. Explain which definition or assumption it violates.
Find its upper endpoint.
The larger zero is 7, and it is included.
A dashed inequality boundary includes every point on its line.
For an upward-opening quadratic, positive values lie outside the roots, not between them. A boundary intersection is not necessarily included. Use a test point not on the boundary itself.
Interpret a new situation
- AQA A22 Higher includes quadratic inequalities in one variable and linear inequalities in two variables. The graphical overlap is the solution region, not one selected point.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the x-coordinate where y=x+1 meets y=5.
Set x+1=5, giving x=4.
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.
The equality line separating values that satisfy an inequality from those that do not. Choose the relationship, show the method, check its assumptions and interpret the result.