Problem 33 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|4 x-5|+3=2$$

Step-by-Step Solution

Answer

The equation has no real solutions, as an absolute value cannot equal a negative number.

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. To do that, subtract 3 from both sides of the equation: \(|4x-5| + 3 - 3 = 2 - 3\) This simplifies to: \(|4x - 5| = -1\)

Step 2: Analyze Results of Absolute Value

The next step is to consider the nature of absolute values. Since the absolute value of an expression is always non-negative, it can never equal a negative number. In this case, we have: \(|4x - 5| = -1\) which is impossible, because an absolute value cannot be -1.

Step 3: Conclude that No Solutions Exist

Since the absolute value cannot equal a negative number, the equation has no real solutions. There are no values of \(x\) that would make \(|4x - 5| = -1\) true.

Key Concepts

Algebra Problem-Solving

When tackling algebra problems, it's essential to follow logical steps to arrive at a solution. Algebra often requires manipulation of equations to isolate variables or expressions that interest us. The goal is to make sense of the equation and simplify it to evaluate the variable at the focus. In this case, our goal was to solve the equation involving the absolute value: \(|4x-5|+3=2\).Here are some strategies to effectively solve algebraic equations:

• Identify like terms and simplify the equation wherever possible.
• Follow the physics of the equation, performing the same operation to both sides to maintain equality.
• Look for patterns or properties specific to the function or expression involved, like absolute values or exponents.

By systematically applying these methods, we transform problems into simpler forms that are easier to solve or analyze. In this case, isolating the absolute value was key to understanding the nature of the equation.

Isolation of Expressions

Isolating expressions is a common step in solving equations. It involves manipulating the equation so that a particular term or expression, usually involving the variable of interest, stands alone on one side. This step profoundly influences how we handle the problem.In the given problem, \(|4x-5|+3=2\), our priority was to isolate the absolute value expression \(|4x-5|\). To accomplish this, we had to eliminate the additional terms around the absolute value. Subtraction of 3 from both sides did just that, transforming the equation into \(|4x-5|=-1\).Successful isolation requires careful operations:

• Identify terms that need to be removed to isolate the expression.
• Apply inverse operations to remove or rearrange these terms.
• Monitor the balance of the equation by applying changes to both sides equally.

With the absolute value isolated, the equation can be examined for further insights, such as understanding whether a valid solution exists.

Non-Negative Properties of Absolute Value

Absolute values are inherently non-negative because they represent the distance between any number and zero on the number line. This property fundamentally changes how they behave in algebraic solutions.In the equation \(|4x-5|=-1\), recognizing the non-negative nature of absolute values reveals an impossibility:

• Absolute values cannot equal a negative number. They are always zero or positive.
• If an equation results in an absolute value set equal to a negative number, there are no real solutions.

This understanding simplifies work involving absolute values as any condition defying non-negativity instantly explains that no solution can exist. Here, knowing that \(|4x-5|=-1\) is unattainable made it clear that the original equation has no real solutions. This insight showcases the importance of familiarizing oneself with the inherent properties of mathematical operations and expressions.