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Algebraic manipulation involves rearranging and simplifying equations or inequalities to make them easier to work with. In our original exercise, we have the inequality \(8x + 3 > 25\). To start solving it, we perform algebraic manipulation by subtracting 3 from both sides. This step helps simplify the inequality to \(8x > 22\).

Subtracting the same quantity from both sides of an equation or inequality ensures that the inequality relationship remains unchanged. This is fundamental because balance is maintained, much like keeping a scale level when weights are removed evenly from each side.

Doing algebraic manipulation, such as addition or subtraction, sets the stage for isolating variables. It is a crucial first step which often appears in solving mathematical expressions.

Variable isolation is the process of manipulating an equation or inequality so that the variable of interest is on one side of the equation or inequality.

In the original problem, once we have our inequality in the form \(8x > 22\), our goal is to isolate \(x\). To do this, we need to remove the 8 that is multiplying the \(x\). We achieve this by dividing both sides of the inequality by 8. This operation results in \(x > \frac{22}{8}\).

It's important to note that division is used here because the 8 is multiplied by the variable. By dividing both sides, we effectively "undo" the multiplication, yielding the expression \(x > 2.75\).

Variable isolation is a core step in solving inequalities, as it helps determine the range of possible values for the variable.

Once you have isolated the variable, you often need to simplify the resulting fractions. Simplifying fractions makes solutions clearer and easier to understand.

After isolating \(x\) to get \(x > \frac{22}{8}\), the next step is simplification. Start by identifying factors common to both the numerator and the denominator. In this case, 22 and 8 are both divisible by 2.

Divide the numerator 22 by 2 to get 11, and the denominator 8 by 2 to get 4. This simplifies our expression to \(x > \frac{11}{4}\). Converting this fraction into a decimal gives \(x > 2.75\).

By simplifying fractions, we get equivalent expressions that are easier to interpret, helping us clearly understand the set of values \(x\) can take.