Problem 25 - Precalculus Student Solutions Manual 5th

Question

The equation \(|7 x+3|=-5 x\) cannot have a positive solution. Why?

Step-by-Step Solution

Answer

|7x + 3| is always non-negative but -5x could be positive or negative, so they can't equal.

Step 1 - Understand the Nature of Absolute Values

The absolute value expression \(|7x + 3|\) represents the distance of \(7x + 3\) from zero, which always yields a non-negative result. That means, \(|7x + 3| \geq 0\) for all real values of \(7x + 3\).

Step 2 - Analyze the Right Side of the Equation

Notice that the right side of the equation is \(-5x\). Depending on the sign of \(x\), \(-5x\) can be positive, zero, or negative.

Step 3 - Find Possibility of Negative Solution

For the absolute value of a number to be equal to \(-5x\), the right side must be non-negative (zero or positive), which contradicts since non-negative cannot be equal to the negative value. Hence, there is no solution that satisfies the given expression.

Key Concepts

Absolute Value Properties

In mathematics, the absolute value of a number refers to its distance from zero on the number line, regardless of its direction. The notation is \(|x|\), where \(|x|\) represents the absolute value of \(\text{x}\). This value is always non-negative.

Some key properties of absolute values include:

• \(|x| \geq 0\) for all real numbers \(x\).
• \(|x|\) is the same as \(x\) if \(x \geq 0\).
• \(|x|\) is \(-x\) if \(x < 0\).

The absolute value reflects the magnitude of a number without consideration for the sign. This means \(|7x + 3|\) will always be non-negative.

Non-Negative Numbers

Non-negative numbers include all positive numbers and zero. They are crucial in understanding absolute value expressions like \(|7x + 3|\).

In this context:

• The term \(|7x + 3|\) represents the distance from zero and, by definition, cannot be negative.
• This equation \(|7x + 3| = -5x\) implies that a non-negative number must equal a potentially negative value on the right side of the equation.

Therefore, it is impossible for a non-negative value, represented by the absolute value expression, to ever equal a strict negative value like \(-5x\).

Equations and Inequalities

Understanding the difference between equations and inequalities is vital. An equation states that two expressions are equal, while an inequality states that one expression is greater or less than another.

When analyzing an absolute value equation like \(|7x + 3| = -5x\), we note:

• The left side is always non-negative due to the nature of absolute values.
• The right side, \(-5x\), can be negative, zero, or positive depending on the value of \(x\).

Given these facts, it’s clear why the original equation has no real solution. The non-negative side (left) can never match a negative value from the right side, creating a fundamental mismatch in the equality.