Without performing any calculations, match the inequality...

Without performing any calculations, match the inequality with its solution. Explain your reasoning.
(a) $2 x \leq-6$
(b) $-2 x \leq 6$
(c) $|x+2| \leq 6$
(d) $|x+2| \geq 6$
(i) $x \leq-8$ or $x \geq 4$
(ii) $x \geq-3$
(iii) $-8 \leq x \leq 4$
(iv) $x \leq-3$

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Key Concepts

Properties of Linear Inequalities

Linear inequalities involve expressions with variables that are compared using inequality signs (such as ? or ?). A key principle is that when both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed. This concept is essential in immediately deducing the solution set of simplified linear inequalities without extensive calculation.

Absolute Value Inequalities

Absolute value inequalities deal with the distance of a number from zero on the number line. They can be split into two types: those that are 'less than or equal to' a number, which represent a closed interval, and those that are 'greater than or equal to' a number, which typically yield two separate ranges. Understanding these forms helps in matching the inequality to the correct solution set by intuitively recognizing the spatial constraints they represent.

Interval Notation

Interval notation is a concise way to represent sets of numbers that satisfy given inequalities. It is particularly useful for expressing the outcomes of absolute value inequalities, where the solutions may form a continuous interval or two distinct intervals. This notation provides clarity in delineating the range or ranges of values that are valid solutions.

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