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Quadratic Inequalities
For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.
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Quadratic Inequalities
What is it?
For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.
In simple words
- For (x-p)(x-q)>0 the product is positive outside the roots (same sign).
- PRODUCT FORM: (x+1)(x-3)>0 when x<-1 or x>3.
- FROM STANDARD FORM: x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10.
- Worked-method reminder: solution: x < -1 or x > 3
Important terms
QUADRATIC INEQUALITIES: For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.
PRODUCT FORM: (x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints are included.
FROM STANDARD FORM: x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10. x^2+4x-21<0 is (x+7)(x-3)<0 so -7<x<3.
What you should learn
- Explain quadratic inequalities using the correct definitions, conditions, and notation.
- Apply outside vs between, test a point correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x<-1 or x>3
- -1<x<9
- Include roots only for <= or >=
PRODUCT FORM
(x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints are included.
FROM STANDARD FORM
x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10. x^2+4x-21<0 is (x+7)(x-3)<0 so -7<x<3.
Recommended for Class 10
Full school-level explanation
For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.
Roots first, then signs:
WHAT — QUADRATIC INEQUALITIES
For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Quadratic Inequalities matters because inequalities describe ranges and limits rather than one exact value. They are used for constraints, graphs, optimisation, and real situations such as budgets, dimensions, and capacity.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through PRODUCT FORM, then FROM STANDARD FORM. Worked: solve (x + 1)(x - 3) > 0. roots: x = -1 and x = 3 intervals: x < -1, -1 < x < 3, x > 3 test x = -2: (-)(-) > 0 true test x = 0: (+)(-) < 0 false test x = 4: (+)(+) > 0 true solution: x < -1 or x > 3 Finish by checking signs, restrictions, units, and whether the answer matches the question.
PRODUCT FORM
(x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints are included.
Remember
- >0 outside if a>0
Examples
- x<-1 or x>3
- -1<x<9
- Include roots only for <= or >=
FROM STANDARD FORM
x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10. x^2+4x-21<0 is (x+7)(x-3)<0 so -7<x<3.
Remember
- factor then chart
Examples
- (x-10)(x+9)>0
- x^2+4x-21<0
- a>0 opens up
Remember
- outside vs between
- test a point
- x < -1 or x > 3
- roots are boundaries
- >0 outside if a>0
- factor then chart
- outside vs between
- test a point
How the ideas connect
Worked: solve (x + 1)(x - 3) > 0. roots: x = -1 and x = 3 intervals: x < -1, -1 < x < 3, x > 3 test x = -2: (-)(-) > 0 true test x = 0: (+)(-) < 0 false test x = 4: (+)(+) > 0 true solution: x < -1 or x > 3
Deeper understanding
Go beyond memorising Quadratic Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve (x + 1)(x - 3) > 0. roots: x = -1 and x = 3 intervals: x < -1, -1 < x < 3, x > 3 test x = -2: (-)(-) > 0 true test x = 0: (+)(-) < 0 false test x = 4: (+)(+) > 0 true solution: x < -1 or x > 3
Common mistakes
Exam focus
Core Idea
PRODUCT FORM
FROM STANDARD FORM
Worked Method
Exam focus
Quick revision
- outside vs between
- test a point
- x < -1 or x > 3
- roots are boundaries
- QUADRATIC INEQUALITIES: For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative betwee
- PRODUCT FORM: (x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints
Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the solution of a strict inequality.
For long answers on Quadratic Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- outside vs between
- test a point
- x < -1 or x > 3
- roots are boundaries
Important formulas
- outside vs between
- test a point
- x < -1 or x > 3
Common exam traps
- Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the solution of a strict inequality.
- Do not treat PRODUCT FORM and FROM STANDARD FORM as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the solution of a strict inequality.
- Mistake
Do not treat PRODUCT FORM and FROM STANDARD FORM as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the solution of a strict inequality.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
outside vs between
test a point
x < -1 or x > 3
roots are boundaries
For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0)
- Definition
- For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0)
- Example
- x<-1 or x>3
- Critical warning
- Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the so
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Exam Practice
Define
Q1. Define Quadratic Inequalities.
Explain
Q2. Explain: PRODUCT FORM.
Example
Q3. Give one clear example of Quadratic Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Quadratic Inequalities.
Exam note
Q5. Write a short exam note on Quadratic Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.