Problem 49 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$3|4-3 x|-4=8$$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = \frac{8}{3} \).

Step 1: Simplify the Equation

Start by isolating the absolute value on one side of the equation. Add 4 to both sides to get:\[ 3|4 - 3x| = 12 \]

Step 2: Divide to Isolate Absolute Value

Divide each side of the equation by 3 to further isolate the absolute value:\[ |4 - 3x| = 4 \]

Step 3: Set Up Two Cases for Absolute Value

The absolute value equation \( |4 - 3x| = 4 \) can be expressed as two separate equations:1. \( 4 - 3x = 4 \)2. \( 4 - 3x = -4 \)

Step 4: Solve the First Case

For the equation \( 4 - 3x = 4 \), subtract 4 from both sides to get:\[ -3x = 0 \]Divide by -3:\[ x = 0 \]

Step 5: Solve the Second Case

For the equation \( 4 - 3x = -4 \), subtract 4 from both sides:\[ -3x = -8 \]Divide by -3:\[ x = \frac{8}{3} \]

Step 6: State the Solution

The solutions to the equation are \( x = 0 \) and \( x = \frac{8}{3} \).

Key Concepts

Understanding Absolute Value Equations

Absolute value equations can often seem daunting at first, but breaking them down makes them quite manageable. The absolute value of a number represents its distance from zero on the number line, regardless of direction. Therefore, solving equations with absolute values involves considering both the negative and positive scenarios.
Imagine you want to solve an equation like \(|a| = b\), where \(b\) is a non-negative number. This equation implies two scenarios:

• \(a = b\)
• \(a = -b\)

Each case must be considered separately. This dual nature is central to the process of solving absolute value equations.
For our original problem, the purpose was to isolate the absolute value term first. Then two separate equations were created, each reflecting one of the potential scenarios based on the value contained within the absolute value sign. Understanding this allows us to solve absolute value equations systematically and effectively.

The Art of Solving Inequalities

Solving inequalities shares similarities with solving regular equations, but with a slight twist. In inequalities, operations affect entire expressions and inequalities can have a range of solutions. Here are the key steps to solving inequalities:

• Isolate the variable on one side of the inequality.
• Perform similar operations as you would in an equation (addition, subtraction, multiplication, division).

One critical point to remember is that if you multiply or divide by a negative number, the inequality sign flips, which requires careful attention.
In solving our example problem involving equalities, the isolated absolute value leads to two equations rather than an inequality directly; however, understanding the principles of inequality helps when these absolute value equations turn into inequalities in other problems. Testing and checking each scenario allows us to determine where the solutions fit on a number line.

Step-by-Step Solutions Demystified

Solving mathematical problems sequentially—step by step—can simplify complex equations into manageable tasks. This is vital for absolute value equations or inequalities, as each step logically leads to the next. Let's look at the approach taken:

• Step 1: Simplify the equation by isolating the absolute value part. Make sure to perform the same operation on both sides to keep the equality balanced.
• Step 2: Break the absolute value equation into two separate cases to solve. Each solution accounts for positive and negative scenarios.
• Step 3: Solve each case individually to ensure no possible solutions are overlooked.

This process ensures systematic analysis, reducing error, and allowing a clearer view of potential solutions. Detailing each step not only builds confidence but also verifies the solution's correctness. Always double-check by plugging values back into the original equations to confirm validity.