Problem 23 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$|5 x-2|=|2-5 x|$$

Step-by-Step Solution

Answer

\(x = \frac{2}{5}\)

Step 1 - Understand the equation

The given equation is \(|5x - 2| = |2 - 5x|\). Notice that the left-hand side is identical to the right-hand side but with terms switched inside the absolute value.

Step 2 - Set up the conditions for the absolute values

For absolute values, examine the cases where the expressions inside may differ. There are two main cases to consider: Case 1: When \(5x - 2 \, \text{and} \, 2 - 5x\) are equal. Case 2: When \(5x - 2\) is equal to the negative of \(2 - 5x\).

Step 3 - Solve Case 1

For Case 1: Assume \(5x - 2 = 2 - 5x\). Solve for \(x\).\(5x - 2 = 2 - 5x \)Add \(5x\) to both sides:\(10x - 2 = 2\)Add 2 to both sides:\(10x = 4\)Finally, divide by 10:\(x = \frac{4}{10} = \frac{2}{5}\).

Step 4 - Solve Case 2

For Case 2: Assume \(5x - 2 = -(2 - 5x)\). Solve for \(x\).\(5x - 2 = -2 + 5x\)Add \(2\) to both sides:\(5x = -4 + 5x\)Subtract \(5x\) from both sides:\(0 = -4\).This case leads to a contradiction, so there is no solution from this case.

Step 5 - Verify the solution

Validate whether \(x = \frac{2}{5}\) satisfies the original equation:When \(x = \frac{2}{5}\), calculate both sides:\(5x - 2 = 5 \, \frac{2}{5} - 2 = 2 - 2 = 0\).\|0| = |0|||.This solution meets the original equation condition.

Key Concepts

Precalculus

Precalculus is a mathematical course that prepares students for calculus. It includes concepts like algebra, trigonometry, and functions.
In precalculus, students learn various ways to solve different types of equations, including equations involving absolute values.
Absolute value equations are particularly important because they express the distance of a number from zero on a number line. This distance measure is always non-negative. Understanding these types of equations is crucial before moving on to more advanced calculus topics.

Solving Equations

Solving equations is a fundamental skill in mathematics. An equation is a statement that shows the equality of two expressions. To solve an equation means to find the value(s) of the variable(s) that make the equation true.
In this exercise, the equation is \( |5x - 2| = |2 - 5x| \). This is an absolute value equation, which involves finding values for \( x \) such that the expression inside the absolute value bars have the same magnitude, regardless of their sign.
To solve this, we consider different cases based on the properties of absolute value.
Here are the general steps to solve absolute value equations:

• Isolate the absolute value expression, if necessary.
• Set up cases based on the definition of absolute value.
• Solve each case for the variable.
• Check each solution in the original equation.

In our problem, we had two cases:

• Case 1: \( 5x - 2 = 2 - 5x \), which simplifies to a valid solution \( x = \frac{2}{5} \)
• Case 2: \( 5x - 2 = -(2 - 5x) \), which results in a contradiction \( 0 = -4 \)

Absolute Value Properties

Absolute value equations rely on the fundamental properties of absolute value. The absolute value of a number \( |a| \) is its distance from zero on the number line. It is always non-negative.
Here are some key properties of absolute values:

• \( |a| \) is always \( \text{≥ 0} \)
• \( |a| = a \) if \( a \) is positive or zero
• \( |a| = -a \) if \( a \) is negative

When solving absolute value equations, these properties allow us to break down the problem into simpler cases. For instance, if \(|A| = |B|\), it implies that

• \( A = B \) or
• \( A = -B \)

The given problem \( |5x - 2| = |2 - 5x| \) can be approached by these properties. We then solve for \( x \) by considering when the expressions inside the absolute values are equal and when they are opposites. This gives us potential solutions, which we then verify to ensure they satisfy the original equation.