Problem 57 - Algebra for College Students

Question

Solve each equation and inequality by inspection. \(|3 x-1|>-2\)

Step-by-Step Solution

Answer

The inequality is true for all real numbers.

Step 1: Understanding Absolute Value

The given inequality is \(|3x - 1| > -2\). Recall that the absolute value of any expression is always non-negative. Hence, \(|3x - 1|\) will always be greater than or equal to zero. Since zero is greater than \(-2\), the inequality \(|3x - 1| > -2\) is always true for any real number \(x\).

Step 2: Conclusion of the Inequality

Since the expression \(|3x - 1|\) is always positive or zero, and \(-2\) is a negative number, the original inequality holds for all real numbers. Thus, there are no restrictions on \(x\), confirming that the inequality is true for all real numbers.

Key Concepts

Exploring Real Numbers

Real numbers are an essential part of mathematics, forming the backbone of many calculations and expressions.Real numbers include all the numbers you frequently encounter:

• Positive and negative whole numbers (integers)
• Fractions (rational numbers)
• Irrational numbers (such as π and √2)
• Zero

Such a comprehensive set of numbers allows us to describe both simple and complex quantities. Real numbers can be easily plotted on a number line, which helps in visualizing inequalities and other expressions.
In the context of our original exercise, we discuss real numbers because the inequality \(|3x - 1| > -2\) holds for 'all' real numbers due to the properties of absolute values.

Solving Inequality Equations

Inequalities are mathematical expressions indicating that values are not strictly equal but instead have a 'less than' or 'greater than' relationship.In our example of \(|3x - 1| > -2\), we deal with an absolute value inequality. It's crucial to recognize how absolute values affect inequality solutions:

• Absolute values, by definition, are never negative. They measure the distance from zero, thus are zero or positive.
• When an expression inside absolute value walls is compared to a negative number, as in our inequality, the solution usually simplifies considerably. Here, \(|3x - 1| \geq 0\) ensures the inequality is valid for any real number, as zero is certainly greater than \(-2\).

The nature of inequalities with absolute values can sometimes lead to universal truths, like in our problem, where it's satisfied by all real numbers.

Understanding Mathematical Expression Evaluation

Evaluating mathematical expressions involves simplifying or resolving them to get an answer or a range of solutions.
In terms of our original exercise, evaluation involves interpreting the absolute value expression \(|3x - 1|\) and its comparison to \(-2\).
Since absolute values always result in non-negative outcomes, this evaluation tells us that \(|3x - 1| > -2\) is inherently true for all values of \(x\). This particular exercise demonstrates a specific scenario where understanding the properties of absolute values simplifies the problem significantly. It highlights the importance of grasping core mathematical concepts for evaluation, enabling quick conclusions in solving inequalities.