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Real numbers are all the numbers you encounter daily. They include whole numbers, fractions, and decimals, both positive and negative. Essentially, real numbers encompass any number that can be located on a number line.

• Whole numbers: like 0, 1, 2, 3…
• Integers: such as -3, 0, 2...
• Rational numbers: these can be fractions like 1/2 or decimals like 0.75
• Irrational numbers: numbers that cannot be expressed as a simple fraction, such as \sqrt{2} \ or \pi \

So when we talk about a variable \(x\) representing real numbers, it can place anywhere on this continuum, not restricted to neat whole numbers or simple fractions.
Understanding real numbers is crucial for solving inequalities because they define the scope of possible solutions. In the context of the inequality \(-80 < x < 0\), \(x\) can be any real number falling between -80 and 0, without including these endpoints.

Strict inequalities are inequalities that do not include the boundary values. They use the symbols \(>\) or \(>\) to convey that a number is strictly greater than or less than another number, without equality.
In the inequality \(-80 < x < 0\), the '<' symbol means x is both greater than -80 and less than 0. It cannot be -80 or 0.This is different from a non-strict inequality, which might use \(\leq\) or \(\geq\) to include boundary values.- **Example:** \(x > 2\) implies x is any number larger than 2, not 2 itself. But \(x \geq 2\) would include 2.A strict inequality defines a range where the values have to be in between two numbers but can't be equal to those numbers themselves.
This is essential when precisely defining a set of numbers, like when setting constraints in mathematical problems.

Number line representation is a visual tool that helps us understand solutions of inequalities more intuitively. By plotting numbers and marking segments on a line, we can easily see which numbers satisfy an inequality.
For the inequality \(-80 < x < 0\), we would draw a horizontal line, place ticks at -80 and 0, and shade the region between these marks without including -80 and 0 themselves. Open circles would typically be placed at -80 and 0 to show these numbers are not included in the solution.- **Drawing tips:** - Place numbers clearly on the line. - Use open circles for excluded boundaries in strict inequalities. - Shade the section that represents solutions to the inequality.This visual tool helps clarify which numbers are viable solutions. It's especially useful for students as they can "see" the scope of possible solutions rather than just interpreting symbolic notation.