Linear Inequalities

• Preservation of Inequality: Adding or subtracting the same value from both sides of an inequality does not change the direction of the inequality symbol. This mirrors the properties of equality used in standard equations.

• The Reversal Principle: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed. This is because multiplying by a negative reflects the values across zero on the number line, effectively swapping their relative order.

• Logical Consistency: If $5 > 3$, multiplying by $-1$ results in $-5$ and $-3$. Since $-3$ is to the right of $-5$ on the number line, the statement must become $-5 < -3$ to remain true.

• Isolation of the Variable: The primary goal is to isolate the variable on one side of the inequality. This is achieved by performing inverse operations in the reverse order of the standard order of operations.

• Rearrangement Strategy: To avoid the risk of forgetting to flip the inequality sign, it is often helpful to rearrange the inequality so that the coefficient of the variable is positive. For example, $10 - 2x > 4$ can be rewritten as $10 - 4 > 2x$.

• Handling Compound Inequalities: When solving simultaneous linear inequalities, solve each one independently and then find the intersection (where both are true) or union (where at least one is true) of the solution sets.

Set Notation

• Uses curly brackets and a vertical bar to define the set: $\{x : x > a\}$, read as 'the set of all $x$ such that $x$ is greater than $a$'.
• Intersection ($\cap$): Represents the region where two inequalities overlap (the 'AND' condition).
• Union ($\cup$): Represents the combined region covered by two inequalities (the 'OR' condition).

Interval Notation

• Uses brackets to define the boundaries of the solution set. Square brackets $[ ]$ indicate that the endpoint is included (inclusive), while round brackets $( )$ indicate it is excluded (exclusive).
• Infinity ($\infty$): Always uses round brackets because infinity is a concept of direction, not a specific reachable value that can be included in the set.

• Strict vs. Weak Inequalities: A strict inequality ($<$ or $>$) excludes the boundary value, represented by an open circle on a number line. A weak inequality ($\leq$ or $\geq$) includes the boundary value, represented by a solid, filled-in circle.

• The Negative Sign Trap: Always double-check if you have multiplied or divided by a negative number. This is the most common source of errors in inequality problems.

• Endpoint Verification: After finding a solution like $x > 5$, test a value slightly larger than 5 (like 6) and a value slightly smaller (like 4) in the original inequality to ensure the direction is correct.

• Infinity Brackets: In interval notation, never use a square bracket for $\infty$ or $-\infty$. This is a technical error that often results in lost marks.

• Reading the Question: Pay close attention to whether the answer must be given in set notation, interval notation, or as a number line graph.