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Rational Equations
State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter.
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Rational Equations
What is it?
State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter.
In simple words
- State excluded values.
- CLEAR: 12x2-16 - 24x-4 = 3.
- CHECK: A solution x=-2 that avoids +/-4 can stand.
- Worked-method reminder: a valid root from the chapter list is x = -2
Important terms
RATIONAL EQUATIONS: State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter.
CLEAR: 12x2-16 - 24x-4 = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.
CHECK: A solution x=-2 that avoids +/-4 can stand. If algebra produced x=4, reject it. Checking in the cleared equation is not enough if you divided information away - check the original.
What you should learn
- Explain rational equations using the correct definitions, conditions, and notation.
- Apply exclude, clear, solve, check correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x != 4, x != -4
- Times LCD
- Get a polynomial
CLEAR
12x2-16 - 24x-4 = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.
CHECK
A solution x=-2 that avoids +/-4 can stand. If algebra produced x=4, reject it. Checking in the cleared equation is not enough if you divided information away - check the original.
Recommended for Class 10
Full school-level explanation
State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter.
Four steps, in order:
WHAT — RATIONAL EQUATIONS
State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Rational Equations matters because rational expressions combine algebra with fraction rules. Factoring, restrictions, and checking are essential because an apparently simple answer may still exclude some values.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through CLEAR, then CHECK. Worked: 12x2 - 16 - 24x - 4 = 3 [outline]. x^2 - 16 = (x-4)(x+4) excluded: x != 4 and x != -4 multiply through by (x-4)(x+4) 12 - 24(x+4) = 3(x-4)(x+4) solve the polynomial check: if a root is +/-4, reject it a valid root from the chapter list is x = -2 Finish by checking signs, restrictions, units, and whether the answer matches the question.
CLEAR
12x2-16 - 24x-4 = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.
Remember
- LCD of all bottoms
Examples
- x != 4, x != -4
- Times LCD
- Get a polynomial
CHECK
A solution x=-2 that avoids +/-4 can stand. If algebra produced x=4, reject it. Checking in the cleared equation is not enough if you divided information away - check the original.
Remember
- original bottoms
Examples
- Reject excluded
- Keep good roots
- Show the substitution
Remember
- exclude, clear, solve, check
- exclude +/-4; x=-2 valid
- check excluded
- LCD of all bottoms
- original bottoms
- exclude, clear, solve, check
- exclude, clear, solve, check
- mathematical reasoning
How the ideas connect
Worked: 12x2 - 16 - 24x - 4 = 3 [outline]. x^2 - 16 = (x-4)(x+4) excluded: x != 4 and x != -4 multiply through by (x-4)(x+4) 12 - 24(x+4) = 3(x-4)(x+4) solve the polynomial check: if a root is +/-4, reject it a valid root from the chapter list is x = -2
Deeper understanding
Go beyond memorising Rational Equations. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: 12x2 - 16 - 24x - 4 = 3 [outline]. x^2 - 16 = (x-4)(x+4) excluded: x != 4 and x != -4 multiply through by (x-4)(x+4) 12 - 24(x+4) = 3(x-4)(x+4) solve the polynomial check: if a root is +/-4, reject it a valid root from the chapter list is x = -2
Common mistakes
Exam focus
Core Idea
CLEAR
CHECK
Worked Method
Exam focus
Quick revision
- exclude, clear, solve, check
- exclude +/-4; x=-2 valid
- check excluded
- LCD of all bottoms
- RATIONAL EQUATIONS: State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (ofte
- CLEAR: 12x2-16 - 24x-4 = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.
Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check.
For long answers on Rational Equations: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- exclude, clear, solve, check
- exclude +/-4; x=-2 valid
- check excluded
- LCD of all bottoms
Important formulas
- exclude, clear, solve, check
- exclude +/-4; x=-2 valid
- check excluded
Common exam traps
- Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check.
- Do not treat CLEAR and CHECK as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check.
- Mistake
Do not treat CLEAR and CHECK 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
Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
exclude, clear, solve, check
exclude +/-4; x=-2 valid
check excluded
LCD of all bottoms
State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original
exclude +/-4; x=-2 valid
- Definition
- State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original
- Formula
- exclude +/-4; x=-2 valid
- Example
- x != 4, x != -4
- Critical warning
- Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check
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Exam Practice
Define
Q1. Define Rational Equations.
Explain
Q2. Explain: CLEAR.
Example
Q3. Give one clear example of Rational Equations and say why it fits.
Mistake
Q4. State one common mistake students make about Rational Equations.
Exam note
Q5. Write a short exam note on Rational Equations: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.