Question

How do you solve an absolute value equation?

Answer 1

The distance between two real numbers $x,y$ is the absolute value of their difference, $|x-y|$. So the statement "all $x$ that are 1/4 away from -6" is captured in the equation

$|x-(-6)|=\dfrac14$

or

$|x+6|=\dfrac14$

To solve, you can use the definition of absolute value: if $x$ is not negative, then $|x|=x$; otherwise, $|x|=-x$.

So there are two cases to consider:

1. Suppose $x+6\ge0$. Then $|x+6|=x+6$, and the equation reduces to

$x+6=\dfrac14\implies x=-\dfrac{23}4$

2. Suppose $x+6$. Then $|x+6|=-(x+6)=-x-6$ and

$-x-6=\dfrac14\implies x=-\dfrac{25}4$

$-6=-\dfrac{24}4$, so the two solutions we found are correct because both $-\dfrac{23}4$ and $-\dfrac{25}4$ are indeed 1/4 away from -6.

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For the question regarding inequality, you know that the absolute value must return a non-negative number. In other words, there is no $x$ such that $|x|$.