MathematicsChapter 4: Linear and Quadratic Inequalities

Fractions, Decimals, All Reals

Clear fractions with a positive LCM.

Learning objectives

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.

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 false, the solution is empty. Identity-looking lines need this check.

Where / symbols

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: (-2/3)x - 2 < (1/3)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.

Overview

Clear, then see if x survives:

Fractions

(-23)x - 2 < (13)x + 8. Times 3: -2x-6 < x+24 -> -3x < 30 -> x > -10 (flip because of / -3).

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.

Examples 1<=4 all reals · 0<=-1 empty · Still check the expansion

WHAT — FRACTIONS, DECIMALS, ALL REALS

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.

Worked example

solve 1 - 8x <= -4(2x - 1).
1 - 8x <= -8x + 4
1 - 8x + 8x <= 4
1 <= 4
this is always true
solution: all real x

In simple words

Expand -4(2x-1) as -8x+4, not -8x-4. The hidden plus is a common wrong empty-set.

Key points

positive LCM
true vs false leftover
times 3 first
x cancelled

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.