Problem 31 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|1.2 x-1.7|-1=3$$

Step-by-Step Solution

Answer

The solutions are \(x = 4.75\) and \(x \approx -1.917\).

Step 1: Isolate the Absolute Value Expression

To solve the equation \(|1.2x - 1.7| - 1 = 3\), we first need to isolate the absolute value expression. Start by adding 1 to both sides of the equation: \(|1.2x - 1.7| = 4\).

Step 2: Set Up Two Separate Equations

The equation \(|1.2x - 1.7| = 4\) implies two possible scenarios: 1. The expression inside the absolute value is equal to 4. 2. The expression inside the absolute value is equal to -4. This gives us two equations to solve: \(1.2x - 1.7 = 4\) and \(1.2x - 1.7 = -4\).

Step 3: Solve the First Equation

Solve the equation \(1.2x - 1.7 = 4\). First, add 1.7 to both sides: \(1.2x = 5.7\). Then, divide both sides by 1.2 to find \(x\): \(x = \frac{5.7}{1.2}\). Calculating this gives \(x = 4.75\).

Step 4: Solve the Second Equation

Next, solve the equation \(1.2x - 1.7 = -4\). Add 1.7 to both sides: \(1.2x = -2.3\). Then, divide both sides by 1.2 to find \(x\): \(x = \frac{-2.3}{1.2}\). Calculating this gives \(x = -1.9167\), approximately.

Step 5: Write the Solutions

The solutions to the original equation are \(x = 4.75\) and \(x = -1.9167\) approximately.

Key Concepts

Solving Equations

When solving equations with absolute values, your aim is to find the values of the variables that make the equation true. The process usually involves several steps.

• First, isolate the absolute value expression. This means getting the absolute value term by itself on one side of the equation.
• Then, break apart the absolute value equation into two cases. Remember, an absolute value equation like \(|A| = B\) leads to two possibilities: either \(A = B\) or \(A = -B\).
• Finally, solve each resulting equation separately to find all possible solutions.

Breaking down the absolute value equation into separate cases is crucial because it reflects the definition of absolute value as a distance, which can be positive in two distinct directions. This method ensures that all potential solutions are identified.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition or multiplication). Understanding the structure of algebraic expressions is key to solving absolute value equations.

• An algebraic expression inside an absolute value, like \(1.2x - 1.7\), involves both a constant and a variable term.
• Manipulating the expression often involves simplifying it by performing operations such as addition or subtraction.
• Then, solving for the variable requires operations that isolate the variable on one side of the equation.

Handling algebraic expressions skillfully is a fundamental skill in algebra. By ensuring variables are properly isolated and constants appropriately handled, students can more easily solve complex equations.

Algebraic Equations

Algebraic equations involve mathematical statements that assert the equality of two expressions. When dealing with absolute value equations, each separate scenario (or equation) derived from the absolute value needs to be solved independently.

• For the given problem, start by setting \(1.2x - 1.7 = 4\) and solving it step by step, beginning with adding the constant to both sides, then isolating \(x\).
• The second equation, \(1.2x - 1.7 = -4\), follows the same process, highlighting that each step involves basic arithmetic operations that maintain equality.
• Solving these algebraic equations requires straightforward calculations, such as division by a coefficient to solve for \(x\).

It's important to solve both scenarios because the absolute value can hold two cases. It reflects the dual nature of absolute values, capturing both positive and negative instances equally.