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Algebraic expressions are combinations of numbers, variables, and operations like addition and subtraction. They form the foundation of algebra and allow us to represent problems in a compact form. For example, the expression \(y - 4\) is an algebraic expression where:

• \(y\) is a variable that can take on different values,
• \(-4\) is a constant that is subtracted from \(y\).

These expressions allow us to create equations and inequalities by equating or comparing them to numbers or other expressions. When manipulating algebraic expressions, our primary goal is often to isolate the variable on one side to solve for its value. In the context of inequalities, this involves performing operations that maintain the integrity of the inequality, such as adding the same number to both sides.

Inequalities are expressions that show the relationship between two values where one is not strictly equal to the other. Instead, they indicate whether one value is greater than, less than, or potentially equal to another. In the problem \(y - 4 < -8\), we are dealing with a strict inequality, indicating that \(y - 4\) is less than \(-8\).

While solving inequalities, similar rules to those of equations apply; you can add, subtract, multiply, or divide both sides of the inequality by the same non-zero value. Remember that multiplying or dividing both sides by a negative number reverses the inequality sign. The solution \(y < -4\) tells us that \(y\) must be any real number that is less than \(-4\).

Graphically, this would mean shading all numbers on a number line to the left of \(-4\), showing the set of solutions.

Real numbers include all the numbers you commonly use in everyday life. They encompass all the rational numbers (like fractions and integers) and irrational numbers (numbers that cannot be expressed as fractions, like \(\sqrt{2}\) or \(\pi\)). In solving inequalities, when we refer to real numbers, we mean any point on the continuous number line that can satisfy the inequality.

Real numbers are important because they provide the widest variety of solutions possible when solving inequalities like \(y < -4\).

So when the problem states that "all real numbers less than \(-4\) are solutions," it implies that this includes anything you can think of that is below \(-4\). This continuous set of numbers answers the inequality, showing the flexibility and breadth of solutions within the realm of real numbers.