Problem 32 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|3-3 x|-2=2$$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{3}\) and \(x = \frac{7}{3}\).

Step 1: Isolate the Absolute Value

First, we need to isolate the absolute value expression. The original equation is \(|3-3x|-2=2\). To do this, add 2 to both sides of the equation which gives us: \(|3-3x|=4\).

Step 2: Set Up Two Equations

The absolute value expression \(|3-3x|\) can be equal to either 4 or -4. This means we must consider two separate cases: 1. \(3-3x = 4\)2. \(3-3x = -4\)

Step 3: Solve the First Equation

For the equation \(3-3x=4\):- Subtract 3 from both sides: \(-3x = 1\)- Divide both sides by -3: \(x = -\frac{1}{3}\)

Step 4: Solve the Second Equation

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

Step 5: Verify Solutions

Substitute \(x = -\frac{1}{3}\) and \(x = \frac{7}{3}\) back into the original equation \(|3-3x|=4\) to verify both satisfy the equation.- For \(x = -\frac{1}{3}\): - \(|3 - 3(-\frac{1}{3})| = |3 + 1| = |4| = 4\)- For \(x = \frac{7}{3}\): - \(|3 - 3(\frac{7}{3})| = |3 - 7| = |-4| = 4\)Both satisfy the equation, so both solutions are correct.

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols usually represent numbers and can be used in formulas to solve numerous problems. In the given problem, we're dealing with an algebraic expression that involves an absolute value. The absolute value sign, represented by vertical bars, shows the distance of a number from zero on a number line, always resulting in a non-negative number.
For example, \([-3\) and \(+3\) both have an absolute value of 3. In our specific problem, \(|3-3x|\), the term inside the absolute value has to be separated from the rest of the equation by isolating it, which is a key algebraic skill. This isolation of terms helps in better tackling and understanding each part of the equation separately.

Equation Solving

Equation solving involves finding a value for the variable that makes the equation true. In our absolute value equation, we isolated the absolute value term to solve for the variable \(x\). Following this, we considered two scenarios due to the nature of absolute values: one where the expression inside the absolute value is equal to positive 4, and another where it equals negative 4.
Both situations were setup as separate equations to solve by applying basic algebraic techniques such as subtraction and division. By doing this, we ended up with two possible solutions for \(x\): \(x = -\frac{1}{3}\) and \(x = \frac{7}{3}\). Each solution requires substitution back into the original absolute value equation to check its validity, ensuring these solutions indeed hold true.

Mathematical Verification

Verification is an essential part of solving mathematical equations as it checks whether the obtained solutions actually satisfy the original equation. For our problem, after obtaining \(x = -\frac{1}{3}\) and \(x = \frac{7}{3}\), both were substituted back into the absolute value expression \(|3-3x|\) to ensure they provide a non-negative result of 4.
This process helps confirm accuracy and reinforces the correctness of solutions found. Without verifying, it's easy to overlook potential mistakes that could arise during manual calculations. This step assures that only the correct solutions are considered as valid.

Mathematical Modeling

Mathematical modeling translates real-world phenomena into mathematical language, using equations like the one we solved to predict and understand various situations. Absolute value equations often represent real-life scenarios where measurements involve distance, which by definition cannot be negative.
For example, in engineering or physics, absolute values can be used to model situations such as stress analysis in materials where both compressive and tensile forces are considered. The skill of mathematical modeling shows how abstract mathematical concepts like absolute value equations can be applied to interpret complex situations in a more structured and understandable manner.