Problem 57 For Problems $55-64$, solve ea... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 2: Problem 57

For Problems $55-64$, solve each equation and inequality by inspection. $$ |3 x-1|>-2 $$

Short Answer

Expert verified

All real numbers $x$ satisfy the inequality.

Step by step solution

Understanding Absolute Value Inequality

Absolute values represent the distance from zero, meaning they are always non-negative. Therefore, any absolute value expression, like $|3x-1|$, is greater than or equal to 0. Thus, it is always greater than $-2$, because any non-negative number is greater than any negative number.

Analyzing the Inequality

The inequality $|3x-1| > -2$ is true for all $x$ because the smallest possible value of $|3x-1|$ is 0, which is still greater than $-2$. Therefore, there is no restriction on the values of $x$.

Conclusion

Since $|3x-1|$ is always greater than $-2$, every real number $x$ is a solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value

Absolute value is a fundamental concept in mathematics that describes the magnitude or distance of a number from zero on the number line.

It is denoted by two vertical bars, like $|a|$, where $a$ is a real number. The absolute value of a number is always non-negative.

This is because it does not consider the direction on the number line, only the distance. For example, $|5| = 5$ and $|-5| = 5$. This property is crucial in understanding inequalities that involve absolute values.

Inequalities

Inequalities are mathematical expressions used to compare the relative sizes or values of two expressions. They show a range of possibilities rather than a specific number.

Common symbols used in inequalities include:

$>$ greater than
$<$ less than
$\geq$ greater than or equal to
$\leq$ less than or equal to

In the context of the exercise, $|3x-1| > -2$ uses an inequality to illustrate a condition that must be satisfied by the expression $|3x-1|$. In this instance, the inequality is always true because the expression $|3x-1|$ cannot be negative, thus always satisfying the inequality by being greater than $-2$.

Real Numbers

Real numbers are essentially all the possible numbers you can think of that can be found on the number line.

They include:

Natural numbers (e.g., 1, 2, 3...)
Whole numbers (e.g., 0, 1, 2, 3...)
Integers (...-2, -1, 0, 1, 2...)
Rational numbers (numbers that can be expressed as a fraction, like 1/2)
Irrational numbers (numbers that cannot be written as simple fractions, like $\sqrt{2}$ or $\pi$)

Real numbers form the basis of many algebraic problems. For the provided inequality $|3x-1| > -2$, every real number $x$ is a solution because the absolute value expression $|3x-1|$ will always be non-negative.

Distance from Zero

Distance from zero on a number line refers to how far a number is from zero, regardless of direction.

In absolute value terms, this distance is always a positive quantity. For any number $a$, its distance from zero is $|a|$.

This means absolute values can be seen as finding out how far away any number is from zero without concerning its sign.

If $a = 7$, then $|a| = 7$ (7 units from zero)
If $a = -7$, then $|a| = 7$ (still 7 units from zero)

Understanding this concept is key when dealing with absolute value inequalities, as they often express a condition based on this distance, ensuring that certain expressions remain within or exceed a particular range based in positive terms.

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91影视

For Problems \(55-64\), solve each equation and inequality by inspection. $$ |3 x-1|>-2 $$

Short Answer

Step by step solution

Understanding Absolute Value Inequality

Analyzing the Inequality

Conclusion

Key Concepts

Absolute Value

Inequalities

Real Numbers

Distance from Zero

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