Problem 12 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$|7-3 x|=3$$

Step-by-Step Solution

Answer

x = \frac{4}{3}\ or x = \frac{10}{3}\

Step 1: Understand Absolute Value

The equation \( |7-3x|=3 \) involves an absolute value. The absolute value of a number is its distance from 0 on the number line, disregarding whether it is positive or negative.

Step 2: Split the Equation

Since the absolute value equation \( |7-3x|=3 \) must account for both positive and negative scenarios, it can be split into two separate equations: \(7-3x=3 \) and \(7-3x=-3 \)

Step 3: Solve the First Equation

First, solve the equation \(7-3x=3\). \ ewline \ 1. Subtract 7 from both sides: \ -3x = 3-7 \ ewline \ 2. Simplify the equation: \ -3x = -4 \ \ ewline \ 3. Divide both sides by -3: \ x = \frac{-4}{-3}\ \ ewline \ 4. Simplify: \ x = \frac{4}{3} \

Step 4: Solve the Second Equation

Next, solve the alternative equation \(7-3x=-3\). \ ewline \ 1. Subtract 7 from both sides: \ -3x = -3-7 \ ewline \ 2. Simplify the equation: \ -3x = -10 \ ewline \ 3. Divide both sides by -3: \ x = \frac{-10}{-3} \ ewline \ 4. Simplify: \ x = \frac{10}{3} \

Key Concepts

absolute value

In mathematics, the absolute value of a number is its distance from zero on the number line. Think of it as the 'size' of the number, ignoring the sign. For example, both \frac{5}\ and \frac{-5}\ have an absolute value of 5. This is denoted using vertical bars like this: \(|5| = 5\) and \(|-5| = 5\). In an equation, dealing with absolute values means considering both positive and negative scenarios for the expression inside the bars. When we have an equation such as \(|7-3x|=3\), we must determine the solutions for both when the expression \(7-3x\) is equal to 3 and when it is equal to -3. This is why absolute value equations always yield two separate cases or solutions, reflecting the dual nature of distances.

splitting equations

When faced with an absolute value equation such as \(|7-3x|=3\), we split the equation into two distinct parts. This is crucial because the absolute value equation can be satisfied in two ways. Thus, splitting the original equation means creating two scenarios: ewline 1) The expression inside the absolute value is equal to 3. \(7-3x = 3\) ewline 2) The expression inside the absolute value is equal to -3. \(7-3x = -3\). ewline By doing this, we convert the problem of solving an absolute value equation into solving two simpler linear equations. This approach guarantees that we explore all possible solutions that satisfy the original absolute value condition.

solving linear equations

Once we have split the absolute value equation into two linear equations, the next step is solving these equations. For example, starting with the first equation: \(7-3x=3\): ewline

• Subtract 7 from both sides: \(-3x = 3-7\)
ewline
• Simplify the equation: \(-3x = -4\)
ewline
• Divide both sides by -3: \(x = \frac{-4}{-3}\)
ewline
• Simplify: \(x = \frac{4}{3}\)

ewline Similarly, for the second equation \(7-3x = -3\): ewline

• Subtract 7 from both sides: \(-3x = -3-7\)
ewline
• Simplify the equation: \(-3x = -10\)
ewline
• Divide both sides by -3: \(x = \frac{-10}{-3}\)
ewline
• Simplify: \(x = \frac{10}{3}\)

ewline By solving these linear equations step-by-step, we can find the values of \(x\) that satisfy both conditions.

positive and negative scenarios

Absolute value equations require us to consider both positive and negative scenarios because of their inherent nature. This dual consideration ensures we account for all values that could potentially satisfy the equation. With our equation \(|7-3x|=3\), we initially have one scenario where the expression inside the absolute value equals a positive 3: \(7-3x = 3\). ewline The second scenario considers the expression inside the absolute value equaling a negative 3: \(7-3x = -3\). ewline This gives us the comprehensive answer set because absolute values measure distance, and distance is always a non-negative quantity. Accounting for both positive and negative scenarios ensures that we don't miss any potential solution.