Explaining the Absence of Real Solutions for |x + 3| = -6

  • MHB
  • Thread starter mathdad
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In summary, the equation |x + 3| = -6 has no real solutions because the absolute value of any number cannot be negative. It is not possible to have imaginary solutions for this equation because the absolute value of a complex number will still result in a positive real value. The graph of this equation would be a horizontal line at y = -6. The absolute value of a negative number is always positive. It is not possible to rewrite this equation to make it solvable because the absolute value of any number will always be positive.
  • #1
mathdad
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Explain why there are no real numbers that
satisfy the equation | x + 3 | = - 6.

I know the reason is because of the negative number. Why is this the case?
 
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  • #2
Let's look at a definition:

\(\displaystyle |u|=\begin{cases}u, & 0\le u \\[3pt] -u, & u<0 \\ \end{cases}\)

Can you see that we must have:

\(\displaystyle 0\le|u|\) ?
 
  • #3
https://mathhelpboards.com/pre-calculus-21/absolute-value-equation-22666.html
 

FAQ: Explaining the Absence of Real Solutions for |x + 3| = -6

Why are there no real solutions for the equation |x + 3| = -6?

The absolute value of any number cannot be negative, so the left side of the equation can never equal a negative number. Therefore, there are no values of x that can make the equation true.

Is it possible to have imaginary solutions for this equation?

No, because the absolute value of a complex number can still only result in a positive real value. Therefore, there are no complex solutions for this equation.

What does the graph of this equation look like?

Since there are no real solutions, the graph of this equation would be a horizontal line at y = -6, indicating that there are no values of x that would make the equation true.

Can the absolute value of a negative number ever be positive?

Yes, the absolute value of a negative number is always positive. For example, the absolute value of -5 is 5.

Is there a way to rewrite this equation to make it solvable?

No, because the absolute value of any number will always result in a positive value. Rewriting the equation would not change this fact and would still result in no real solutions.

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