Problem 14 - Algebra for College Students

Question

Solve each equation. $$ |5 x-1|=21 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{22}{5}\) and \(x = -4\).

Step 1: Understand Absolute Value

The absolute value of a number ensures that the result is always non-negative. Therefore, if \(|a| = b\), then it translates to two potential equations: \(a = b\) and \(a = -b\).

Step 2: Set Up Two Equations

Given \(|5x - 1| = 21\), set up the two potential equations:1. \(5x - 1 = 21\)2. \(5x - 1 = -21\)

Step 3: Solve the First Equation

Solve \(5x - 1 = 21\):Add 1 to both sides to get \(5x = 22\).Divide both sides by 5: \(x = \frac{22}{5}\).

Step 4: Solve the Second Equation

Solve \(5x - 1 = -21\):Add 1 to both sides to get \(5x = -20\).Divide both sides by 5: \(x = -4\).

Step 5: Write the Solution Set

Combine the solutions from both equations to write the full solution set: \(x = \frac{22}{5}\) and \(x = -4\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is key in solving absolute value equations. The absolute value of a number is essentially its distance from zero on the number line, regardless of direction. This means that the absolute value is always non-negative. To understand, think of \(|a| = b\). This tells us there are two possible scenarios: either \(a = b\) or \(a = -b\). For example, \(|5| = 5\) and \(|-5| = 5\). This property is crucial when breaking down absolute value equations into simpler linear equations.

Solving Linear Equations

Once we understand absolute values, the next step is simplifying them into linear equations. When we start with \(|5x - 1| = 21\), we can translate it into two separate equations due to the definition of absolute value:
- \(5x - 1 = 21\)
- \(5x - 1 = -21\)

Both equations are linear because they follow the format \(ax + b = c\), where \(x\) is the variable we solve for. Solving these linear equations involves straightforward algebra:

• **First Equation:** Add 1 to both sides getting \(5x = 22\). Now, divide by 5 like so: \(x = \frac{22}{5}\)
• **Second Equation:** Add 1 to both sides getting \(5x = -20\). Now, divide by 5 like so: \(x = -4\)

Linear equations are simpler and help isolate the variable so we can determine specific values for \(x\).

Understanding the Solution Set

The solution set of an equation is the collection of all values of the variable that make the equation true. For the equation \(|5x - 1| = 21\), we have found two solutions: \(x = \frac{22}{5}\) and \(x = -4\). Each of these solutions is valid because, when plugged back into the original absolute value equation, they satisfy the condition \(|5x - 1| = 21\).

Therefore, the solution set for this equation is \(\bigg\{\frac{22}{5}, -4\bigg\}\). This signifies that the equation holds true for both values of \(x\). Always make sure to combine all found solutions to write the complete solution set accurately.