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When we talk about numbers in algebra, we often focus on the set of real numbers. The real numbers include all the numbers on the number line, which contains rational numbers (like 6, \frac{2}{3}, and -3.75), irrational numbers (like \(\sqrt{2}\) and \(\pi\)), integers, whole numbers, and natural numbers.

Real numbers can be represented graphically on a number line, which helps us to understand inequalities better. Values to the left on this line indicate smaller numbers, while values to the right indicate larger numbers. Using this visual aid enables us to grasp how numbers compare to one another, an essential skill when working with inequalities.

Algebraic inequalities are statements that describe the relative size or order of two values. They are used to show that one quantity is smaller, larger, or equal to another quantity. The most common inequality symbols are \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to).

Inequality notation is a shorthand way to express an interval or range of real numbers that satisfy the inequality. For instance, \( x < 5 \) signifies that x can be any real number less than 5 but not including 5 itself. Understanding and using the correct notation is crucial in communicating mathematical ideas clearly and accurately.

Solving inequalities involves finding all the possible values of the variable that make the inequality true. Just like equations, the goal is to isolate the variable on one side. However, unlike equations, when multiplying or dividing by a negative number, the inequality sign must be flipped.

Let's apply this concept to our example, where we have \( t \leq 20 \). This inequality already shows the solution—any real number for \( t \) that is less than or equal to 20 is part of the solution set. If we were to solve a more complex inequality, such as \( 3t + 10 > 40 \), we would subtract 10 from both sides and then divide by 3, remembering to maintain the direction of the inequality unless we divide by a negative.

The solution to inequalities is often visualized on a number line, where we can highlight or shade the section representing all the possible values that satisfy the inequality.