Problem 4 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, solve for \(x\). $$ |4+3 x|=1 $$

Step-by-Step Solution

Answer

The solutions are \(x = -1\) and \(x = -5/3\).

Step 1: Set Up Absolute Value Equation

The equation is \(|4 + 3x| = 1\), meaning the expression inside the absolute value can be either positive or negative. Therefore, there are two separate equations to consider: \(4 + 3x = 1\) and \(4 + 3x = -1\).

Step 2: Solve the Positive Case

Solve the equation \(4 + 3x = 1\):\Subtract 4 from both sides:\3x = 1 - 4\Simplify:\3x = -3\Divide both sides by 3:\x = -1

Step 3: Solve the Negative Case

Solve the equation \(4 + 3x = -1\):\Subtract 4 from both sides:\3x = -1 - 4\Simplify:\3x = -5\Divide both sides by 3:\x = -5/3

Step 4: Combine Solutions

The solutions to the original equation are \(x = -1\) and \(x = -5/3\).

Key Concepts

absolute value

Understanding absolute value is crucial when solving equations like \(|4 + 3x| = 1\). The absolute value of a number represents its distance from zero on a number line, regardless of direction. So, \(|-3| = 3\) and \(|3| = 3\). Absolute value expressions can be split into two separate linear equations to account for both positive and negative scenarios.
In our exercise, \ |4 + 3x| = 1 \ becomes two equations: \4 + 3x = 1\ and \4 + 3x = -1\.
To solve these equations, we handle each case separately.

linear equations

Linear equations are equations where the variable, in this case, \( x \), is raised only to the power of 1. They take the general form \( ax + b = c \).
In our specific problem, we have two linear equations to solve:

• 4 + 3x = 1
• 4 + 3x = -1

By solving these, we will find all possible values of \(x\).

solving equations

To solve the equations, we follow a systematic approach. Let's start with \(4 + 3x = 1\).

• Subtract 4 from both sides: \(3x = 1 - 4\).
• This simplifies to \(3x = -3\).
• Finally, divide by 3: \(x = -1\).

Next, solve the second equation, \(4 + 3x = -1\).

• Subtract 4 from both sides: \(3x = -1 - 4\).
• This simplifies to \(3x = -5\).
• Finally, divide by 3: \(x = -5/3\).

By combining these steps, we find both solutions for the original equation: \(x = -1\) and \(x = -5/3\).