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When we talk about real numbers, we're referring to the numbers you can find on a typical number line. This includes all the numbers you're familiar with, like whole numbers (1, 2, 3, ...), fractions (like 1/2, 2/3), irrational numbers (numbers like \( \pi \) or \( \sqrt{2} \) that cannot be expressed as a simple fraction), and negative numbers (like -1, -2, -3, ...). In fact, any number you can think of that doesn't involve imaginary components (like the square root of a negative number) is a real number.

Real numbers have certain properties that make them a fundamental concept in mathematics. They can be added, subtracted, multiplied, and divided (except for division by zero), and they follow the commutative, associative, and distributive properties. When graphing sets of real numbers, understanding the types of numbers within the set is crucial to correctly displaying them on a number line.

The number line representation is a visual tool used to depict numbers and their relationships. Imagining or drawing a straight, horizontal line, we can place numbers along it in order, often using ticks or marks. We traditionally place 0 in the middle, with positive numbers to the right and negative numbers to the left.

To

plot a single number

, you simply draw a dot at the position on the number line corresponding to that number's value. For intervals or sets of numbers, like \( x \leq 7 \), we take additional steps. In this case, placing a solid dot on 7 signifies that 7 is included in the set. Then, we draw a line or ray extending leftward to cover all numbers less than 7. If 7 wasn't included, we'd use an open circle to represent this exclusion.

The process of solving inequalities is similar to solving equations, with the key difference involving the range of possible answers. An inequality like \( x \leq 7 \) does not say 'x is 7', but rather 'x is any real number up to and including 7'. So, there isn't just one solution, there's a whole set of numbers that satisfy the inequality.

To represent the solutions to an inequality graphically on a number line, interpret the inequality symbol. For \( x \leq 7 \), all numbers less than or equal to 7 work. So, just like in your step-by-step solution, you would plot a shaded line or ray extending from 7 towards increasingly smaller numbers, and include a filled-in or solid circle at 7 to indicate that 7 itself is a solution. This visual representation helps you quickly see the range of possible values for x. When dealing with inequalities, always remember to consider whether the number corresponding to the inequality symbol is included (solid dot) or not (open circle).