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Chapter 1: Problem 14

Solve the equation. $$ 20+|2 x-4|=15 $$

Short Answer

Expert verified
The equation has no solution because an absolute value cannot be negative.

Step by step solution

01

Isolate the Absolute Value Expression

Start by moving the 20 from the left side of the equation to the right side, by subtracting 20 from both sides.\[20 + |2x - 4| - 20 = 15 - 20\]This simplifies to:\[|2x - 4| = -5\]
02

Recognize a No-Solution Scenario

The absolute value of any expression is always non-negative (zero or positive). Since it is stated that \(|2x - 4| = -5\), which implies an absolute value equals a negative number (-5), there is no possible solution for this equation as no absolute value can ever be negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

No-Solution Scenario
An absolute value equation can sometimes lead to a scenario where there is no solution. This happens when the equation forces the absolute value to equal a negative number.
For example, consider the equation \(|2x - 4| = -5\). Absolute values represent the distance of a number from zero on the number line, and distances are always non-negative.
Therefore, an expression like \(|2x - 4| = -5\) where an absolute value equals a negative number has no solution. Keep these points in mind:
  • Absolute values can never be negative.
  • If solving an equation results in an absolute value equating to a negative number, it indicates a no-solution scenario.
  • Always double-check for errors before concluding a no-solution scenario to ensure the initial equation was interpreted correctly.
Isolation of Absolute Value
Isolating the absolute value is a key step to solving these types of equations. It involves manipulating the equation so that the absolute value expression is alone on one side.
In our example \( 20 + |2x - 4| = 15 \), we need to move the 20 to the other side by subtracting it from both sides.
This simplifies the equation to \(|2x - 4| = -5\). Important strategies for isolating the absolute value include:
  • Use basic arithmetic operations such as addition and subtraction.
  • Perform these operations on both sides of the equation to maintain balance.
  • Ensure the absolute value is completely isolated before proceeding to solve it further or check for solutions.
Properties of Absolute Value
Understanding the properties of absolute value is crucial for solving related equations and recognizing no-solution scenarios. Absolute value measures the magnitude or distance of a number from zero. Here are some key properties:
  • Non-negative: Absolute values are always \( \geq 0 \). They are never negative.
  • Symmetrical: \(|a| = |-a|\) for any real number \(a\). Distance from zero is the same in both negative and positive directions.
  • Zero is Zero: \(|0| = 0\), because zero is the point of reference.
Understanding these properties helps avoid any errors when solving equations. It also immediately highlights impossible scenarios, such as trying to equal a negative value with an absolute value expression.

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