Problem 11 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \((-\infty,-7]\)

Step-by-Step Solution

Answer

The interval \((- fty, -7]\) is expressed as the inequality \(x \leq -7\).

Step 1: Understand the Interval Notation

The given interval is \((-\infty, -7]\), which means it includes all real numbers less than or equal to -7. \((-,\infty\) signifies that there is no lower boundary, and \(-7]\) means the interval includes -7.

Step 2: Convert to Inequality

To express this interval as an inequality, we represent it as: \[ x \leq -7 \]. The inequality captures all values of \(x\) that are less than or equal to -7, continuing from negative infinity.

Key Concepts

interval notation

Interval notation is a mathematical method used to denote a range of values, and it's particularly important in algebra when dealing with inequalities. This notation tells us which numbers are included in a set and offers clarity on endpoints.
Brackets are used in interval notation to show whether endpoints are included or excluded:

• Square brackets \( [\) and \( ]\) indicate that the endpoint is included in the interval. For example, \( [a, b]\) includes both \( a\) and \( b\).
• Parentheses \( (\) and \( )\) indicate that the endpoint is not included. For example, \( (a, b)\) does not include \( a\) or \( b\).

In the given exercise, the interval \( (-\infty, -7]\) was converted to an inequality. The interval starts from negative infinity (which cannot be included as it's a concept, not a number) and ends at -7. Here, \( ]\) implies that -7 is included in the interval.

real numbers

Real numbers encompass all the numbers on the number line, including both rational numbers (such as fractions and integers) and irrational numbers (like \( \pi \) or \( \sqrt{2} \)). They form the basic set of numbers used in algebra.

• Rational numbers include fractions, as well as negative and positive whole numbers and zero.
• Irrational numbers are non-repeating, non-terminating decimals, and cannot be expressed as a simple fraction.

Real numbers are integral to understanding intervals, as they encompass all possible values within those intervals. In our interval \( (-\infty, -7]\), we're dealing with real numbers that are less than or equal to -7. These include all numbers on the number line from negative infinity up to and including -7.

inequalities in algebra

In algebra, inequalities are statements about the relative size or order of two values. They use symbols to express these relationships, with common symbols being:

• \( > \): greater than
• \( < \): less than
• \( \geq \): greater than or equal to
• \( \leq \): less than or equal to

Converting intervals into inequalities is a fundamental task in algebra, as it allows us to work with ranges of numbers. For example, the interval \( (-\infty, -7]\) was expressed as an inequality \( x \leq -7 \). This inequality tells us that \( x \) can be any number that is less than or equal to -7.
Algebraic inequalities are utilized to solve problems involving unknown variables, analyze mathematical conditions, and even in real-world applications where constraints need to be defined.