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When solving linear inequalities, such as \( -9x \geq 36 \), it's important to understand the steps involved in manipulating the inequality to isolate the variable. Initially, one should identify the inequality sign and the terms that are intertwined with the variable. In this case, \( -9 \) is multiplied with \( x \) and \( 36 \) is the constant on the other side of the inequality.

Inequality manipulation involves performing operations that maintain the balance of the inequality. This includes adding, subtracting, multiplying, or dividing both sides by the same positive number. However, exercise caution when multiplying or dividing by negative numbers, as this will impact the inequality sign – a concept we'll discuss in depth under 'inequality symbol reversal'.

Our goal in manipulating inequality is to simplify the expression to a point where the variable, in this case, \( x \), stands alone on one side of the inequality, clearly showing its relationship with the other terms.

The process of isolating variables is a key step in solving linear inequalities. By 'isolating', we mean rearranging the inequality so that the variable we're solving for, such as \( x \), is by itself on one side of the inequality. This step makes it easier to identify the set of possible values that \( x \) can take.

In the given exercise, we aim to isolate \( x \) from \( -9x \geq 36 \). To do this, we want to undo the multiplication of \( -9 \) and \( x \). We achieve this by performing the opposite operation, which is division. When we divide both sides of the inequality by \( -9 \) (the coefficient of \( x \) ), we edge closer to finding the value of \( x \). It's essential to perform the same operation on both sides to maintain the equality's integrity.

Remember, we do this to ensure single-variable terms like \( x \) are evident, giving us a clearer understanding of the variable's possible values in relation to the inequality.

A crucial concept in solving inequalities is the 'inequality symbol reversal'. When both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. This is a rule that is fundamental to maintaining the truth of the inequality.

In our exercise, when we divide \( -9x \geq 36 \) by \( -9 \) to isolate \( x \), we must flip the inequality sign. Therefore, the greater than or equal to sign (\geq) becomes less than or equal to sign (\leq). As a result, \( x \leq -4 \) becomes our simplified inequality. Understanding when and why to reverse the sign is essential; it ensures the correct representation of the variable's relationship to other numbers.

This reversal occurs because dividing by a negative number is akin to multiplying all terms by \( -1 \) and flipping the inequality's direction. For example, if \( a < b \) is true, then \( -a > -b \) will also be true due to the nature of negative numbers in relation to each other.