Graphing Absolute Value Equations and Inequalities as Systems

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The point (8,2.4) is on the graph. Using these three points, two function rules can be written. One for the decreasing part, and the other for the increasing part of the graph. The domain of these parts can be written as follows. Decreasing Part:& 0 ≤ x < 4 Increasing Part:& 4 ≤ x ≤ 8 Note the point at x=4 can belong to either piece, so long as it belongs to only one of them. The next step will then be to find the equation for both lines. For this, recall the slope-intercept form of a linear function. y= mx + b In this form m is the slope of the line and b is the y-intercept. Start with the decreasing part. The y-intercept and slope can be found from the graph.

For this line the y-intercept is b= 2.4 and the slope is m= - 0.6. Decreasing Part & Domain y= - 0.6 x + 2.4 & 0 ≤ x < 4 Similarly, an equation for the increasing part can also be written.

For the increasing line, the y-intercept is - 2.4 and the slope is 0.6. Increasing Part & Domain y= 0.6 x - 2.4 & 4 ≤ x ≤ 8 Knowing the equations for both lines, the absolute value function can be written as a piecewise function. f(x)= - 0.6x+2.4 & if 0 ≤ x < 4 0.6x-2.4 & if 4 ≤ x ≤ 8

f(x)= - 0.6x+2.4 & if 0 ≤ x < 4 0.6x-2.4 & if 4 ≤ x ≤ 8 To do so, the function rules can be rearranged so that one rule contains an expression and the other rule contains its opposite.

The function rules contain the expressions - (x-4) and x-4. These expressions produce non-negative values in their domains. Therefore, they can be written using absolute values. f(x)= - 0.6(x-4) & if 0 ≤ x < 4 0.6(x-4) & if 4 ≤ x ≤ 8 ⇓ f(x) = 0.6|x-4| The domain of this absolute value function is the union of the domains for the function rules. Function: & f(x) = 0.6|x-4| Domain: & 0 ≤ x ≤ 8

h(x) = - 0.6 |x-200|+120 ⇕ h(x) = - 0.6( - [x-200])+120 & if x < 200 - 0.6( x-200)+120 & if x ≥ 200 The function rules can be simplified.

Next, each function rule will be drawn separately and their graphs will be combined. First draw the graph of y=0.6x.

The graph ends with an open circle because its domain is the set of x-values less than 200. For x-values greater than or equal to 200, the graph of y=- 0.6x+240 will be drawn. This is a linear function written in slope-intercept form. y = - 0.6x + 240 ⇓ l Slope: - 0.6 y-intercept: 240 Using this information, its graph can be drawn.

This piece ends with a closed circle, as 200 is in its domain. Finally, both graphs will be combined on the same coordinate plane.

Since all the measures are in meters, the distance between these two points is 400 meters.

g(x - 1) ≥ 5|x - 1-0.5| - 1.5 ⇓ y ≥ 5|x-1.5| - 1.5 Considering the definition of absolute values, this inequality can be divided into two pieces. y ≥ 5 |x-1.5| - 1.5 ⇓ y ≥ - 5 [ - (x - 1.5)] - 1.5 & if x - 1.5 < 0 5( x - 1.5) - 1.5 & if x - 1.5 ≥ 0 The expressions can be simplified.

Now, the inequalities describing the domains of the pieces will be rearranged. To do so, 1.5 will be added to both sides of the inequalities. y ≥ - 5x+6 & if x<1.5 5x-9 & if x≥ 1.5

The boundary line is in slope-intercept form. Its graph can be drawn by using the slope and y-intercept. Since the inequality is not strict, the boundary line will be solid. Additionally, since x is less than 1.5, the boundary line will have an open endpoint at x=1.5.

To determine the region to be shaded, choose a test point and substitute it into the inequality — for example, (0,0).

Since the point does not satisfy the inequality, the region that does not contain the test point will be shaded. Note that only points in the corresponding domain x<1.5 will be considered.

Next, the second piece of the inequality will be graphed.

The boundary line's slope is 5. Also, (1.5,- 1.5) satisfies the second piece. This point is the leftmost point of the boundary line because its domain is x≥ 1.5. Additionally, since the inequality is non-strict, the boundary line will be solid.

Now, choose a point not on the boundary line whose x-coordinate is greater than or equal to 1.5 — for example, (4,0).

Since the point did not satisfy the inequality, the region that does not contain the point that will be shaded.

Finally, the graphs of the two pieces can be combined on the same coordinate plane.

Maya can now complete her image.

External credits: Hari Panicker