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Chapter 3: Problem 31

Solve. $$7-|2 x-1|=6$$

Short Answer

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Step by step solution

01

- Isolate the absolute value

Start by isolating the absolute value expression on one side of the equation. Subtract 7 from both sides of the equation: 7 - |2x - 1| = 6 |-|2x - 1| = 6 - 7 |-|2x - 1| = -1
02

- Recognize the contradiction

Recall that the absolute value function, |a|, always results in a non-negative number. However, the equation |- |2x -1| = -1 presents a scenario where the absolute value equals a negative number, which is impossible.

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Key Concepts

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

Isolating absolute value
When solving an absolute value equation, the first step is to isolate the absolute value expression. This means getting the expression inside the absolute value brackets alone on one side of the equation.
For example, in the equation 7 - |2x - 1| = 6, by subtracting 7 from both sides we get:
|-|2x - 1| = 6 - 7
|-|2x - 1| = -1.
Now, we have isolated the absolute value expression. This is a crucial step because it allows us to focus solely on the absolute value part of the equation.
Non-negative results
The absolute value of a number is always non-negative. This means that for any value inside the absolute value brackets, the outcome will be zero or positive.
Mathematically, for any real number a, |a| is always greater than or equal to zero.
By understanding this property, we can quickly identify if an equation involving absolute values is feasible. For example, with the isolated expression |2x - 1| = -1,
we see that the right side is negative. This situation is impossible since the absolute value can never equal a negative number.
Equation contradiction
If isolating the absolute value yields a scenario where the absolute value equals a negative number, then the equation has no solution. This is termed as an 'equation contradiction'.
Let's revisit our example: 7 - |2x - 1| = 6
Isolating the absolute value, we got |2x - 1| = -1.
Since |2x - 1| can never be -1 (as absolute values are always non-negative), we face a contradiction. Thus, the equation has no possible solution.
Always check for contradictions when solving absolute value equations because they can help you quickly determine if an equation has a solution or not.

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