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Chapter 0: Problem 73

Explain why the equation $$|8 x-3|=-2$$ has no solutions.

Short Answer

Expert verified
The equation \(|8x-3| = -2\) has no solutions because the absolute value of any expression is always non-negative, meaning it is either zero or positive. It is impossible for the absolute value function \(|8x-3|\) to be equal to a negative number like -2.

Step by step solution

01

Understanding the Absolute Value Function

Before solving the given equation \(|8x-3| = -2\), let's understand the absolute value function. The absolute value function is represented by the symbol "||" and, by definition, it is the distance between a number and zero on the number line. Therefore, the absolute value of any number is always non-negative, meaning it is either zero or positive. For example, \(|5| = 5\) and \(|-5| = 5\).
02

Analyzing the Given Equation

Now let's examine the given equation \(|8x-3| = -2\). The absolute value of any expression will always be non-negative, so the equation asks us to find a value of x for which the absolute value of an expression is equal to a negative number (-2).
03

Explaining the Impossibility of Solutions

As we have established that the absolute value of any expression is always non-negative, it is impossible for the absolute value function \(|8x-3|\) to be equal to a negative number like -2. Therefore, the given equation \(|8x-3| = -2\) has no solutions.

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Key Concepts

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

Absolute Value Function
Place your html The absolute value of any number can be understood as its distance from zero on a number line, regardless of direction. This distance is always expressed as a non-negative number because it represents the magnitude without considering the direction. For instance, both 5 and -5 are five units away from zero, making their absolute values identical, \( |5| = 5 \). This property of always being non-negative is crucial when solving equations that include absolute values.
No Solution
An equation may sometimes lead to a scenario where no valid answers exist that satisfy the equation, and we describe this as 'no solution.' In the case of absolute value equations, this occurs when the equation sets the absolute value equal to a negative number. Since the definition of absolute value constrains it to non-negative values only, any equation stipulating it to be negative, such as \( |8x-3| = -2 \), is inherently unsolvable. Recognizing situations that yield no solution is essential for correctly analyzing mathematical problems.
Non-negative Numbers
Non-negative numbers are precisely what their name suggests: numbers that are not negative. They comprise all positive numbers, including zero. In other words, they are all the numbers on the number line to the right of zero, as well as zero itself. They are significant in the context of absolute values because the outcome of an absolute value function will always be a non-negative number. This set includes whole numbers, fractions, decimals, and zero鈥攂ut excludes any number with a negative sign in front of it.
Number Line
A number line is a visual representation of numbers laid out in a straight line where each point corresponds to a number. It is an essential tool in understanding the concept of absolute values, as it provides a spatial interpretation of the value鈥檚 magnitude. On a number line, zero is the central point, with positive numbers extending to the right and negative numbers to the left. The distance of a number from zero, or its absolute value, is represented by the length of a line segment from zero to the number, disregarding the direction.

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Most popular questions from this chapter

Write each union as a single interval. $$(-\infty,-10] \cup(-\infty,-8]$$

Find all numbers \(x\) satisfying the given inequality. $$\frac{x-2}{3 x+1}<2$$

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(-3, \infty) \cap[-5, \infty)$$

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Show that $$|| a|-| b|| \leq|a-b|$$ for all real numbers \(a\) and \(b\).
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