Problem 18 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ 20+|2 x-4|=15 $$

Step-by-Step Solution

Answer

The equation has no solution because \(|2x - 4|\) cannot be negative.

Step 1: Isolate the Absolute Value

To solve the equation \(20 + |2x - 4| = 15\), we first need to isolate the absolute value. Subtract 20 from both sides of the equation: \[|2x - 4| = 15 - 20\]Thus, the equation simplifies to:\[|2x - 4| = -5\]

Step 2: Analyze the Absolute Value Equation

The expression \(|2x - 4| = -5\) suggests the magnitude of \(2x - 4\) should equal \(-5\). However, the absolute value of any real number cannot be negative.

Step 3: Conclude the Solution

Since an absolute value cannot be negative, the equation \(|2x - 4| = -5\) has no solution.

Key Concepts

Solving Equations

In mathematics, solving equations is all about finding the values of unknown variables that make the equation true. An equation is a statement showing two expressions are equal.
When solving, the goal is to isolate the variable of interest, commonly represented by letters like \(x\), and determine its value.
Equations could be simple linear forms such as \(ax + b = c\), or they can be more complex, involving polynomials or other functions like absolute values.

• Standard Steps: Rearrange the equation by performing mathematical operations like addition, subtraction, multiplication, or division on both sides.
• Checking your Solution: Substitute the values back into the original equation to verify the accuracy.

When dealing with absolute values, as seen in the example \(20 + |2x - 4| = 15\), the goal is to handle the absolute value expression separately to solve the equation effectively.

Isolation of Absolute Value

Isolating the absolute value in an equation is a critical step when dealing with absolute value expressions. The absolute value of a number or expression represents its non-negative magnitude, shown by two vertical bars.
In equations, isolating this expression assists in examining its value more effectively.
For the equation \(20 + |2x - 4| = 15\), the process involved subtracting 20 from both sides:

• The operation gives us \(|2x - 4| = -5\).

This reveals the importance of isolating the absolute value, as having the isolated expression is crucial to understanding and solving the problem.

Now, the isolated equation \(|2x - 4| = -5\) must be assessed for possible solutions.

No Solution Cases

Sometimes, certain equations have no valid solutions. This often occurs when the solution contradicts fundamental mathematical principles.
In the context of absolute values, a no solution case arises when the absolute value is set equal to a negative number.
Absolute values, by definition, are never negative. They represent the distance from zero and are always zero or positive.

The equation \(|2x - 4| = -5\) is such a case., as it asks for a negative which is impossible.

In these scenarios, the conclusion is straightforward; no numbers exist to satisfy the equation.
It’s essential to recognize these cases early to avoid unnecessary calculations and confusion.

Being aware of such conditions helps identify the characteristics of equations and improves problem-solving skills. Understanding why no solutions exist in particular cases hones a student's intuition for when to apply specific strategies.