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Understanding and manipulating algebraic expressions are crucial skills in solving various types of mathematical problems. An algebraic expression is a mathematical phrase containing numbers, variables, and operation symbols.

Take, for example, the expression from our exercise \(2x-11+3(x+2)\). It involves variables (\(x\)), constants (\(-11\) and \(2\)), and arithmetic operations (addition and multiplication). Simplifying such an expression requires combining like terms and distributing any factors over terms within parentheses.

In our specific problem, simplifying starts with distributing the \(3\) across the terms within the parentheses, giving us \(3x+6\). We then combine like terms by adding or subtracting the coefficients of \(x\) and grouping constant terms. The result is a simpler, equivalent expression, which makes further manipulations, such as solving inequalities, more straightforward.

When dealing with functions, function simplification is a process similar to simplifying algebraic expressions, but it also sets the stage for understanding more complex function properties.

The goal is to rewrite the function in the simplest form possible without changing its values. This usually means combining like terms and reducing the expression to a core representation that is easier to evaluate. For the function \(f(x) = 2x - 11 + 3(x+2)\), simplifying requires us to distribute the multiplication over the addition and then combine the \(x\) terms and the constants.

After simplification, our function becomes \(f(x) = 5x - 5\). This form is much simpler to work with, especially when we're trying to understand how the function behaves, such as what inputs give non-positive outputs. It's essential to simplify functions before attempting to solve or graph them.

Inequality solutions involve finding the values for the variables that make the inequality true. Inequalities can be thought of as balance scales — when solving them, we seek to maintain balance by performing the same operations on both sides.

In our exercise, we were tasked to solve the inequality \(5x - 5 \leq 0\). To do this, we aim to isolate \(x\) on one side of the inequality. Adding \(5\) to both sides of the inequality eliminates the \( -5\), resulting in \(5x \leq 5\). This means that the scale balances when \(5x\) is at most \(5\). Dividing both sides by \(5\) then gives us the most simplified form of the inequality: \(x \leq 1\).

All values of \(x\) less than or equal to \(1\) will keep the balance scaled in favor of the inequality being true. This forms the solution set, which is the range of values that satisfy the original inequality. Understanding how to find and represent the inequality solutions is vital in both pure and applied mathematics, including fields like economics and engineering.