Problem 49 - Elementary Algebra

Question

Determine the inequality given the answers expressed in interval notation. $$ (-\infty, 7] $$

Step-by-Step Solution

Answer

The inequality is \(x \leq 7\).

Step 1: Understand Interval Notation

Interval notation \((-\infty, 7]\) describes all numbers from negative infinity up to and including 7.

Step 2: Translate to Inequality Form

The interval \((-\infty, 7]\) translates to the inequality \(x \leq 7\). This means that \(x\) can be any real number less than or equal to 7.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions showing the relationship between two values. They tell us if one value is greater than, less than, or possibly equal to another. When you see symbols like \(<\), \(>\), \(\leq\), or \(\geq\), they indicate inequalities.
For example:

• \(x < 5\) means \(x\) is less than 5.
• \(y \geq -2\) means \(y\) is greater than or equal to -2.

With inequalities, we define ranges for variables, helping us understand all possible values they can take. This makes inequalities a powerful tool for solving real-world problems, such as finding acceptable ranges for quantities or prices. Allies in our math journey, they let us set boundaries and describe the space where certain conditions hold true.

Real Numbers

Real numbers are all the numbers that you can find on the number line. They include virtually all numbers you use in everyday life. Among them, you'll find:

• Positive numbers, like 1, 2, and 3.
• Negative numbers, like -1, -2, and -3.
• Zero.
• Fractions, like \(\frac{1}{2}\) and \(-\frac{3}{4}\).
• Irrational numbers like \(\pi\) (pi) and \(\sqrt{2}\).

The set of real numbers combines rational and irrational numbers, providing a comprehensive system for measurement and calculation.
Why is this important in inequalities? Because when we express inequalities, we're often dealing with real numbers, determining the limits and boundaries within this expansive and inclusive set.

Translating Interval Notation to Inequality

Interval notation provides a shorthand way to express a range of values. Instead of describing which numbers are included one by one, intervals neatly condense this information. For example, the interval notation \((-fty, 7]\) describes all real numbers less than or equal to 7.
To translate this into an inequality, follow the symbols:

• \((-\infty, 7]\) uses a round parenthesis \(()\) before \(-\infty\), indicating that infinity is a concept and not a number, so it can never be included.
• The square bracket \([]\) after 7 indicates that 7 is part of the set, meaning the values include exactly 7.

This is expressed in inequality form as \(x \leq 7\), where \(x\) represents any number in the range fulfilling the condition. Understanding this translation helps bridge the visual representation in math with the logical conditions they represent.