Absolute Value Equations

[lhwalker]

Students are required to be able to solve systems of equations that may include absolute value functions A-REI.11.  However, no where in the Standards do I see that students need to solve an absolute value equation, for example |2x + 3| = 10.  Is this correct?

[Cathy Kessel]

Giving some examples in the order in which the associated standards appear (which is not necessarily the order in which associated abilities might be learned).

In A-REI, under the cluster heading: Represent and solve equations and inequalities graphically
 

11. Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions. (emphasis added)

This might be done for f(x) = |2x + 3| and g(x) = 10, finding an approximate solution to f(x) = g(x), that is |2x + 3| = 10, via technology as in the standard, although I’d hope that students could do it by hand. 

 
Some other situations that might involve solving |2x + 3| = 10 below.
 
In F-IF:
 

Analyze functions using different representations

7. Graph functions expressed symbolically and show key features of

the graph, by hand in simple cases and using technology for more

complicated cases.

b. Graph square root, cube root, and piecewise-defined functions,

including step functions and absolute value functions. (emphasis added)

A key feature of f(x) = |2x + 3| – 10 is where it intersects the x-axis. To find this by hand (assuming it’s a simple case), solve 0 = |2x + 3| – 10. 
 
Or, understand its graph as a translation of f(x) = |2x + 3|. Students learn about translations in Grade 8, in particular that they preserve length, and this is continued for graphs of functions in F-BF:
 
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),
f(kx), and f(x + k) for specific values of k (both positive and negative);
find the value of k given the graphs. Experiment with cases and
illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and
algebraic expressions for them.
 

For example, take f(x) = |x|.

Graph f(2x), f(2x + 3), f(2x + 3) – 10, noting x-intercepts of each.

[lhwalker]

Thank you for your quick response.   In the interest of narrowing curriculum, it makes sense to have the expectation that students solve 5-4|2x-3| = -11 as a system of equations.  After all, we would not teach a separate method for finding solutions for something like cos x – 3 tan x = 7.

For 5-4|2x-3| = -11, our high school students struggle with having to first use order of operations to get the absolute value alone:  |2x-3| = 4  , then take the bars off and write 2x-3 =4 or 2x-3 = -4, and then solve for x twice.  It all becomes a confused, memorized procedure.     On the other hand, if students know 6.NS.7c (the absolute value of a rational number is its distance from zer0 on the number line), the solutions to |x| = 3 can be visualized by writing x= above both the 3 and the -3 on a number line, then it would not be such a leap for them to write 2x-3 on the number line twice, once above 4 and once above negative 4.

[Cathy Kessel]

I wasn’t meaning to suggest that students work only with the number line (as opposed to graphs in the plane). But if they do, a teacher might use an unmarked length (e.g., a ruler with its marks not showing) or one of those large compasses for classroom use to illustrate the location of the two places on the number line that are 3 units from the origin.

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