Absolute Value Equations and Graphs: Practice

Every number has a distance from zero, so nothing is excluded.

$\text{domain: all real } x$

An absolute value is never negative, and it reaches $0$ at the vertex.

$y \ge 0$

The value $0$ is included, so the range uses $\ge$, not a strict $>$.

Two numbers sit exactly six units from zero, one on each side.

$x = 6 \quad \text{or} \quad x = -6$

The only number at distance zero from zero is zero itself.

$x = 0$

A right side of $0$ gives exactly one solution, not two.

An absolute value is never negative, so it cannot equal $-4$.

$|x| \ge 0, \text{ which can never equal } -4$

Because the right side is negative, there is no solution.

On negative inputs the rule negates the number, and negating a negative gives a positive.

$x < 0 \;\Rightarrow\; |x| = -x$

For example $|-5| = -(-5) = 5$.

A constant added outside the bars raises every output.

$y = |x| + 2 \text{ shifts the graph up } 2$

Subtracting $3$ outside the bars lowers the vertex from $(0, 0)$.

$(0, 0 - 3) = (0, -3)$

Compute inside the bars first, then take the absolute value.

$|7 - 10| = |-3| = 3$

Compare each number's distance from zero.

$|-9| = 9 > |7| = 7 > |-5| = 5 > |3| = 3$

The farthest from zero is $-9$.

The vertex sits where the inside of the bars is zero.

$x - 4 = 0 \;\Rightarrow\; x = 4, \quad y = |0| = 0$

So the vertex is $(4, 0)$.

By definition it is a distance, so it is zero or positive.

$|x| \ge 0 \text{ for every real } x$

The smallest it gets is $0$, at $x = 0$.