Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |7 x+2|>-2 $$

Step-by-Step Solution

Answer

The inequality \(|7x + 2| > -2\) holds for all real numbers: \((- finity, + finity)\).

Step 1: Understanding Absolute Inequality

The problem involves an inequality with an absolute value. The inequality is given by \(|7x + 2| > -2\).

Step 2: Analyzing Absolute Value Properties

The expression \(|7x + 2| > -2\) indicates that the absolute value is greater than -2. However, the absolute value of any real number is always greater than or equal to zero.

Step 3: Conclusion About the Inequality

Since \(|7x + 2|\) is always non-negative (\(\geq 0\)), it is always greater than -2. Therefore, the inequality is satisfied for all real numbers.

Step 4: Express the Solution in Interval Notation

Since the inequality holds true for all real values of \(x\), the solution is \((-finity, +finity)\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities can be a bit tricky, but once you understand the essential properties of absolute values, they become manageable. The absolute value of a number or expression, like \(|7x + 2|\), represents its distance from zero on the number line, irrespective of direction. Because of this, it is always non-negative. Therefore, when you come across an inequality such as \(|7x + 2| > -2\), it may initially seem complex, but remember this key concept: - The absolute value of anything is always greater than or equal to zero. Hence, any statement claiming that an absolute value is greater than a negative number is always true. The nature of absolute values takes precedence in determining the solution, often simplifying possible solutions considerably. As in this case, where the absolute value condition is true for any real number, making the inequality hold for all values of \(x\). Understanding this concept can immensely simplify the process of solving similar expressions.

Interval Notation

Interval notation is a simplified way of expressing a range of numbers, and it comes in handy when dealing with solutions to inequalities. It is particularly beneficial because it provides a clear and standardized format that simplifies understanding and communication of numerical ranges. In the context of our solved inequality \(|7x + 2| > -2\), we discovered that all real numbers satisfy this condition. Therefore, the solution extends across the entire number line. In interval notation, this is expressed as \((-\infty, +\infty)\).

• Parentheses \(()\) indicate that the endpoints are not included in the set.
• Brackets \([]\) can be used to include endpoints when needed, but not in this case as \(-\infty\) and \(+\infty\) are concepts, not distinct values.

Always ensure you write the lower bound on the left. Given the nature of inequality solutions, mastering interval notation is essential for clearly conveying results.

Real Numbers

Real numbers include all the values on the number line that you can think of; they encompass rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). In the context of inequalities, especially those involving absolute values, the real number set plays an important role. In solving the inequality \(|7x + 2| > -2\), the key realization is that because the absolute value is always non-negative, it does not limit \(x\). Consequently, every real number is a solution. Real numbers are comprehensive and essential in mathematics because they fill every gap on the number line, making them continuous without any holes or jumps.

• The set of real numbers, \(( -\infty, +\infty )\), is infinite and unbounded, meaning it stretches endlessly in both positive and negative directions.
• In interval notation, when you write \((-\infty, +\infty)\), it signifies all possible real numbers.

Understanding and visualizing real numbers and their properties is crucial for resolving inequalities and representing solutions comprehensively.