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Chapter 1: Problem 27

Solve the equation. $$3|x+1|-5=-11$$

Short Answer

Expert verified
No solution: absolute values can't equal negative numbers.

Step by step solution

01

Isolate the absolute value expression

Start with the equation \(3|x+1|-5=-11\). Add 5 to both sides to isolate the absolute value term: \[3|x+1|-5+5=-11+5\] Simplifying gives: \[3|x+1|=-6\]
02

Divide by the coefficient of the absolute value

Divide both sides by 3 to solve for the absolute value expression: \[\frac{3|x+1|}{3}=\frac{-6}{3}\] This simplifies to: \[|x+1|=-2\]
03

Analyze the absolute value equation

The expression \(|x+1|=-2\) is invalid because the absolute value is always non-negative (zero or positive). There are no real solutions where an absolute value equals a negative number.

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Key Concepts

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

Isolate Absolute Value
When solving an equation with an absolute value, the first and crucial step is to isolate the absolute value expression.
Think of it as peeling away layers to reach the center of an onion: each operation simplifies the equation until you're left with the absolute value by itself.
In our equation, we start with \(3|x+1|-5=-11\).
  • First, we add 5 to both sides. This balances the equation and removes the subtracted 5: \(3|x+1|-5+5=-11+5\).
  • After simplifying, you'll get: \(3|x+1|=-6\).
  • Finally, to isolate \(|x+1|\), divide each side by 3: \(\frac{3|x+1|}{3}=\frac{-6}{3}\). This leaves us with \(|x+1|=-2\).
By isolating the \(|x+1|\), we made the equation ready for solving if it weren't for the negative result.
No Real Solutions
Understanding that absolute values are non-negative is key to figuring out if an equation has real solutions.
Absolute values measure distance, and distance can never be negative.
Once we reach \(|x+1|=-2\), this signals a problem.
  • The absolute value \(|x+1|\) must be zero or a positive number because it's the distance of \(x+1\) from zero.
  • An equation stating \(|x+1|=-2\) implies that 'distance' is negative, which doesn't make sense in real numbers.
Therefore, when attempting to solve the equation, we realize it has no real solutions, as an absolute value cannot equal a negative number.
This insight helps avoid errors in real-life problem-solving scenarios.
Negative Absolute Value
A negative absolute value in an equation serves as an instant alert that there's no possible solution.
This is because absolute values are defined by their non-negative output.
  • If you find yourself simplifying an equation to something like \(|expression| = negative\), you immediately know it's incorrect in terms of real solutions.
  • In mathematical terms: \(|\text{anything}|\)+ something non-negative \( \geq 0\).
So, when faced with \(|x+1| = -2\), remember:- The equation indicates nothing can make \(x+1\) reduce its absolute value to a negative.- There's no real solution for this equation, saving you time and effort trying to solve what mathematically isn't solvable.This concept protects against misinterpretations and guides learners to recognize unsolvable problems quickly.

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