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Fractions, Decimals, All Reals
Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always false, the solution is empty. Identity-looking lines need this check.
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Fractions, Decimals, All Reals
What is it?
Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always false, the solution is empty. Identity-looking lines need this check.
In simple words
- Clear fractions with a positive LCM.
- FRACTIONS: (-23)x - 2 < (13)x + 8.
- ALL x OR NONE: 1-8x <= -4(2x-1) -> 1-8x <= -8x+4 -> 1<=4, true for all x.
- Worked-method reminder: solution: all real x
Important terms
FRACTIONS, DECIMALS, ALL REALS: Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always false, the solution is empty. Identity-looking lines need this check.
FRACTIONS: (-23)x - 2 < (13)x + 8. Times 3: -2x-6 < x+24 -> -3x < 30 -> x > -10 (flip because of / -3).
ALL x OR NONE: 1-8x <= -4(2x-1) -> 1-8x <= -8x+4 -> 1<=4, true for all x. A contradiction leftover means no x.
What you should learn
- Explain fractions, decimals, all reals using the correct definitions, conditions, and notation.
- Apply positive LCM, true vs false leftover correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x > -10
- Flip only at the -3
- Keep every term
FRACTIONS
(-23)x - 2 < (13)x + 8. Times 3: -2x-6 < x+24 -> -3x < 30 -> x > -10 (flip because of / -3).
ALL x OR NONE
1-8x <= -4(2x-1) -> 1-8x <= -8x+4 -> 1<=4, true for all x. A contradiction leftover means no x.
Recommended for Class 10
Full school-level explanation
Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always false, the solution is empty. Identity-looking lines need this check.
Clear, then see if x survives:
WHAT — FRACTIONS, DECIMALS, ALL REALS
Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always false, the solution is empty. Identity-looking lines need this check. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Fractions, Decimals, All Reals 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 FRACTIONS, then ALL x OR NONE. Worked: solve 1 - 8x <= -4(2x - 1). 1 - 8x <= -8x + 4 1 - 8x + 8x <= 4 1 <= 4 this is always true solution: all real x Finish by checking signs, restrictions, units, and whether the answer matches the question.
FRACTIONS
(-23)x - 2 < (13)x + 8. Times 3: -2x-6 < x+24 -> -3x < 30 -> x > -10 (flip because of / -3).
Remember
- times 3 first
Examples
- x > -10
- Flip only at the -3
- Keep every term
ALL x OR NONE
1-8x <= -4(2x-1) -> 1-8x <= -8x+4 -> 1<=4, true for all x. A contradiction leftover means no x.
Remember
- x cancelled
Examples
- 1<=4 all reals
- 0<=-1 empty
- Still check the expansion
Remember
- positive LCM
- true vs false leftover
- all real x
- distribute the minus
- times 3 first
- x cancelled
- positive LCM
- true vs false leftover
How the ideas connect
Worked: solve 1 - 8x <= -4(2x - 1). 1 - 8x <= -8x + 4 1 - 8x + 8x <= 4 1 <= 4 this is always true solution: all real x
Deeper understanding
Go beyond memorising Fractions, Decimals, All Reals. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve 1 - 8x <= -4(2x - 1). 1 - 8x <= -8x + 4 1 - 8x + 8x <= 4 1 <= 4 this is always true solution: all real x
Common mistakes
Exam focus
Core Idea
FRACTIONS
ALL x OR NONE
Worked Method
Exam focus
Quick revision
- positive LCM
- true vs false leftover
- all real x
- distribute the minus
- FRACTIONS, DECIMALS, ALL REALS: Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is alwa
- FRACTIONS: (-23)x - 2 < (13)x + 8. Times 3: -2x-6 < x+24 -> -3x < 30 -> x > -10 (flip because of / -3).
Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.
For long answers on Fractions, Decimals, All Reals: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- positive LCM
- true vs false leftover
- all real x
- distribute the minus
Important formulas
- positive LCM
- true vs false leftover
- all real x
Common exam traps
- Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.
- Do not treat FRACTIONS and ALL x OR NONE as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.
- Mistake
Do not treat FRACTIONS and ALL x OR NONE 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
Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
positive LCM
true vs false leftover
all real x
distribute the minus
Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always f
- Definition
- Clear fractions with a positive LCM. If the inequality reduces to a statement with no x that is always true, the solution is all real numbers. If it is always f
- Example
- x > -10
- Critical warning
- Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.
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Exam Practice
Define
Q1. Define Fractions, Decimals, All Reals.
Explain
Q2. Explain: FRACTIONS.
Example
Q3. Give one clear example of Fractions, Decimals, All Reals and say why it fits.
Mistake
Q4. State one common mistake students make about Fractions, Decimals, All Reals.
Exam note
Q5. Write a short exam note on Fractions, Decimals, All Reals: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.