Problem 45 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|6-2 x|+1=3$$

Step-by-Step Solution

Answer

x = 2 or x = 4

Step 1: Isolate the Absolute Value Expression

Subtract 1 from both sides of the equation to isolate the absolute value expression: \[ |6-2x| + 1 - 1 = 3 - 1 \]This simplifies to:\[ |6-2x| = 2 \]

Step 2: Set Up Two Separate Equations

The absolute value equation \(|6-2x| = 2\) splits into two separate equations:1. \(6-2x = 2\)2. \(6-2x = -2\)

Step 3: Solve the First Equation

Solve the first equation:\[ 6-2x = 2 \]Subtract 6 from both sides:\[ -2x = -4 \]Finally, divide both sides by -2:\[ x = 2 \]

Step 4: Solve the Second Equation

Solve the second equation:\[ 6-2x = -2 \]Subtract 6 from both sides:\[ -2x = -8 \]Finally, divide both sides by -2:\[ x = 4 \]

Step 5: Check Your Solutions

Substitute the solutions back into the original equation to verify:For \(x=2\):\[ |6 - 2(2)| + 1 = 3 \]\[ |6 - 4| + 1 = 3 \]\[ |2| + 1 = 3 \]\[ 2 + 1 = 3 \]This is true.For \(x=4\):\[ |6 - 2(4)| + 1 = 3 \]\[ |6 - 8| + 1 = 3 \]\[ |-2| + 1 = 3 \]\[ 2 + 1 = 3 \]This is also true.

Key Concepts

solve absolute value equations

When you come across an absolute value equation, it may look intimidating at first. However, it can be broken down into a few straightforward steps. Absolute value equations, like \(|6-2 x|+1=3\), involve finding solutions for a variable within absolute value brackets that result in a given value. Here's a general approach to solving these types of equations:

• Identify the absolute value expression.
• Isolate the absolute value expression (if necessary).
• Set up two separate equations: one for the positive value and one for the negative value.

Breaking it down, we start by isolating \(|6-2x|\) from the original equation \(|6-2 x|+1=3\). This involves subtracting 1 from both sides to get \(|6-2 x|=2\).
Next, we set up two new equations to address both possible scenarios of the absolute value: \[ 6-2x = 2 \] and \[ 6-2x = -2 \] By solving each equation individually, we find the possible values of \(x\). Solving \(6-2x=2\), we get \(x=2\). Then, solving \(6-2x=-2\), we find \(x=4\). These steps ensure we comprehensively address both possible values inside the absolute value brackets.

isolate absolute value expressions

Before solving an absolute value equation, it is essential to isolate the absolute value expression. This simplifies the equation and allows you to focus on solving within the absolute value brackets. Here's a detailed look into this process using our example:
In the equation \(|6-2 x|+1=3\), the goal is to isolate \(|6-2 x|\). We achieve this by performing inverse operations on both sides of the equation. First, subtract 1 from both sides to remove the constant term: \(|6-2x|+1-1=3-1\)
This simplifies to: \(|6-2 x|=2\), giving us a clean equation with the absolute value isolated.
Isolating the absolute value expression is crucial because it prepares the equation for the next step, which is solving for the variable within the absolute value brackets. Remember, the goal is always to have the absolute value expression alone on one side of the equation before creating the two separate linear equations needed to find the solutions.

checking solutions in equations

Verification of the solutions is a critical step to ensure accuracy. After solving the equations derived from the absolute value inequality, you must substitute the solutions back into the original equation to confirm they satisfy it. For our example:

• First, solve for \(x=2\):
  Substitute \(x=2\) back into the original equation \(|6-2 x|+1=3\):
  \(|6-2(2)|+1 =3\) simplifies to \(|2|+1=3\), which is true since \(2+1=3\).
• Next, solve for \(x=4\):
  Substitute \(x=4\) back into the original equation:
  \(|6-2(4)|+1 =3\) simplifies to \(|-2|+1=3\), which is true because \(2+1=3\).

This verification step ensures that the solutions \(x=2\) and \(x=4\) are correct. Not only does it provide confidence in your results, but it also helps identify any possible mistakes. Always remember to check your solutions in the original equation as a final step.