## Solve This #1 – Wrap Up

This was pretty interesting. I posted a blog asking for solutions to a problem, and got a lot of feedback.

Colleen Young (@ColleenYoung on Twitter) in the comments & Colin Graham (@ColinTGraham on Twitter) on Twitter used a graphical approach. They graphed $y=|x^2-9|$ and noted where the function was between 2 & 9.

David Radcliffe (@daveinstpaul on Twitter) and Liz Durkin (@LizDK on Twitter) both used an algebraic approach that was right on the money.

Shawn Urban (@stefras on Twitter) used a unique algebraic approach as shown in the blog post Algebra 2: Solving Absolute Value Equations by Kate Nowak (@k8nowak on Twitter). I hadn’t really seen that before, and liked it a great deal. I may work it in next semester when solving absolute value equations/inequalities.

My Solution

What jumped out to me was a graphical approach as well. I thought about graphing the parabola $y=x^2-9$, and then determining where the function was between 2 & 9 and between -9 & -2.

All that remains at that point is finding the endpoints of the interval algebraically.

• $x^2-9=9$
• $x^2-9=2$
• $x^2-9=-2$
• $x^2-9=-9$

Of course, I imagine many students might initially miss the problem at x = 0.

Second Solution

Another approach that came to me was to initially solve the inequality $|x^2-9|<9$, and then taking out the solutions to $|x^2-9|<2$. These two inequalities are quite easy to solve algebraically as they are intersections as opposed to unions.

In other words, I’d start with $(-\sqrt {18},0) U (0, \sqrt{18})$ and then delete the intervals $(-\sqrt{11},-\sqrt{7})$ and $(\sqrt{7},\sqrt{11})$.

Summary

I think this can be a fun feature of the blog. I will try to work it in on a regular basis. If you have any ideas for problems to include, let me know. I also welcome comments on my two approaches to this problem.

– George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.