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Inequality manipulation refers to the various algebraic operations we perform to find the solution range for an algebraic inequality. An essential skill in algebra, inequality manipulation involves steps like adding, subtracting, multiplying, dividing, and applying reverse operations to maintain the balance of inequality throughout the manipulation process.

Take, for example, the given exercise \( -5 \leq 6x - 12 \). To begin with, we need to 'undo' the operations performed on the variable \(x\). Since \(x\) has been multiplied by 6 and then 12 has been subtracted from it, we reverse these operations. This strategy requires us to add 12 on both sides, which makes the inequality \(7 \leq 6x\) after simplification. It is essential to remember that the inequality sign does not change when we add or subtract a number from both sides.

It's also crucial to know when the inequality direction changes: this happens when we multiply or divide by a negative number. Since all the numbers we work with here are positive or zero, the direction of the inequality remains the same.

Understanding the Tools

Algebraic operations form the toolkit for solving equations and inequalities alike. Familiar operations such as addition, subtraction, multiplication, and division are the bread and butter of algebra. When applied with precision, these operations help unravel the variables and lead us to the solution.

Referring back to our exercise, after adding 12 to both sides to eliminate the -12, the second step is to divide both sides by 6, the coefficient of \(x\). This is a standard operation in algebra to isolate the variable. In the inequality \(7 \leq 6x\), we divide by 6 to get \(\frac{7}{6} \leq x\), which simplifies the inequality for \(x\). We divide all terms by the same number to maintain equality, which is why \(6x/6 = x\). This move ensures that the isolation of the variable is done without altering the inequality's true meaning.

When we talk about isolating variables, we're describing the process of manipulating algebraic expressions to get the variable alone on one side of the equation or inequality. It's a foundational technique that is not only used in solving equations but is crucial in inequalities as well.

In our example, the goal is to isolate \(x\) to find the values it can take that satisfy the inequality. The final step after the previous algebraic operations is to ensure that the variable stands alone. When we divide both sides of \(7 \leq 6x\) by 6, we're effectively peeling away the numerical coefficient that's been attached to \(x\), leaving us with \(\frac{7}{6} \leq x\) or \(x \geq \frac{7}{6}\). This final form not only isolates \(x\) but also presents the solution in the simplest fashion, demonstrating the range of values that \(x\) can take and thus solving the original inequality.