Graphing Absolute Value Inequalities

The equation will be written in vertex form as follows. It is now in vertex form, and its vertex is $(\text{-}2,4).$ Since the $x\text{-}$coordinate of the vertex is $\text{-}2,$ a table of values will be made using some $x\text{-}$values below.

$x$	$\text{-}\frac{5}{2}|x+2|+4$	$y=\text{-}\frac{5}{2}|x+2|+4$
$\text{-}4$	$\text{-}\frac{5}{2}|\text{-}4+2|+4=\text{-}1$	$\text{-}1$
$\text{-}3$	$\text{-}\frac{5}{2}|\text{-}3+2|+4=1.5$	$1.5$
$\text{-}2$	$\text{-}\frac{5}{2}|\text{-}2+2|+4=4$	$4$
$\text{-}1$	$\text{-}\frac{5}{2}|\text{-}1+2|+4=1.5$	$1.5$
$0$	$\text{-}\frac{5}{2}|0+2|+4=\text{-}1$	$\text{-}1$

Finally, plot the points and draw the absolute value equation.

As can be seen, the mirror has a V-shape with a vertex of $(\text{-}2,4)$ in the coordinate plane.

The shaded region represents the $y\text{-}$values greater than $\text{-}\left|\frac{5}{2}x+5\right|+4.$ The inequality symbol will be strict because the graph of the mirror is not part of the solution set. In this example, the mirror represents the boundary line of the inequality, the spherical light source represents a test point that satisfies the inequality, and the shaded region represents the graph of the inequality.

If Ignacio can find someone who is ready to pay greater than or equal to the amount after $x$ years, the object is sold. Therefore, the following inequality will describe the situation. To graph this absolute value inequality, its boundary line will be drawn first. To do so, make a table of values for $y=0.4|x-4|+3.$ Note that since $x$ represents the number of years, it cannot be negative.

$x$	$y=0.4|x-4|+3$	$y$
$0$	$y=0.4|0-4|+3$	$4.6$
$2$	$y=0.4|2-4|+3$	$3.8$
$4$	$y=0.4|4-4|+3$	$3$
$6$	$y=0.4|6-4|+3$	$3.8$
$8$	$y=0.4|8-4|+3$	$4.6$

Plot the points and draw the boundary line. Since the inequality is non-strict, the boundary line will be solid. Note that only positive values of $x$ and $y$ makes sense in the context of the situation.

Next, the region to be shaded will be determined. To do so, choose an arbitrary point not on the boundary line and substitute it into the inequality. Let the point be $(0,0).$ Since $0\ge 4.6$ is a false statement, it is not a solution to the inequality. Therefore, the region that does not contain the test point should be shaded. This means that Ignacio will be willing to sell the object at any price greater than or equal to $\$5000.$ Therefore, $\$5000$ is the minimum price Ignacio can accept after $9$ years.