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exploring_inequalities.tex
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exploring_inequalities.tex
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% In 2020, 4 out of 16 students thought that a<7 implies that a<=7 is not true.
% In 2020, 3 out of 16 students did not generate counterexamples with negatives.
\yourname
\activitytitle{Exploring inequalities}{}
\vspace*{-0.2in}
\overview{In this activity you will explore properties of inequalities without proving the inequalities.
You will use examples and counterexamples to sharpen your intuition about inequalities and their properties.
Be adventurous when you look for counterexamples; don't forget negative numbers.
If you find a counterexample, put a box around it.
If the conclusion seems to be correct, put a big check mark next to it.}
\question{Is the statement $7 \leq 7$ true? Explain.}{-0.1in}
\question{Is the statement $7 < 7$ true? Explain.}{-0.1in}
\question{Is the statement $6 \leq 7$ true? Explain.}{-0.1in}
\question{Suppose $a < 7$. Can you conclude that $a \leq 7$? This is counterintuitive for many people.
Writing $a \leq 7$ does not mean, ``I certify that $a$ really could equal 7.'' Instead, it means ``$a<7$ or $a=7$''. Alternatively, ``$a > 7$ is false.'' Again, is it also true that $a \leq 7$?}{0.0in}
\question{Suppose $a \leq 7$. Can you conclude that $a < 7$? A good technique is to write down five numbers satisfying $a \leq 7$ and see if they also satisfy $a < 7$. Try to find a counterexample. If you find one, put a box around it, otherwise put a check mark.}{0.0in}
\question{Suppose $a < b$ and $b \leq c$. Is it guaranteed that $a \leq c$? Work with examples, and look for a counterexample. One concrete counterexample would be enough, but don't stop at just one positive example.}{0.0in}
\question{Suppose $a < b$ and $b \leq c$. Is it guaranteed that $a < c$? Work with examples, and look for a counterexample. One concrete counterexample would be enough, but don't stop at just one positive example.}{0.0in}
\question{Suppose $a \leq b$ and $b \leq c$. Is it guaranteed that $a < c$? Work with examples, and look for a counterexample. One concrete counterexample would be enough, but don't stop at just one positive example.}{0.0in}
\question{Suppose $a > 12$. Consider the inequality $-a > -12$. Write down five numbers satisfying $a > 12$ and check whether or not they satisfy $-a > -12$. Look for a counterexample. If you find a counterexample, put a box around it. If the result seems OK, put a check mark.}{0.1in}
\question{Suppose $a > 12$. Consider the inequality $-a < -12$. Write down five numbers satisfying $a > 12$ and check whether or not they satisfy $-a < -12$. If you find a counterexample, put a box around it.}{0.0in}
\question{Suppose $c < 5$. Use examples to check whether $c^2 < 25$. If you find a counterexample, put a box around it and look for an additional condition on $c$ which would guarantee $c^2 < 25$.}{0.0in}
\pagebreak
\question{Suppose $c < 7$ and $d \leq 8$.
Use examples to check whether $c + d < 15$.
If you find a counterexample, put a box around it and look for an additional condition on $c$ and $d$ which would guarantee $c + d < 15$.}{0.1in}
\question{Suppose $c < 3$ and $d \leq 4$.
Use examples to check whether $cd < 12$.
If you find a counterexample, put a box around it and look for an additional condition on $c$ and $d$ which would guarantee $cd < 12$.}{0.1in}
\question{Suppose $a < b$ and $c > 0$.
Use examples to check whether $ac < bc$, as above.
}{0.1in}
\question{Suppose $a \leq b$ and $c \leq d$.
Use examples to check whether $a+c < b+d$, as above.
If you find a counterexample, box it and look for an additional condition to guarantee $a+c < b+d$.}{0.1in}
\question{Suppose $a < b$ and $c \leq d$.
Use examples to check whether $ac < bd$.
If you find a counterexample, put a box around it and look for an additional condition which would guarantee $ac < bd$.}{0.2in}
\question{Suppose $a \leq b$.
Use examples to check whether $a^2 < b^2$.
If you find a counterexample, put a box around it and look for an additional condition which would guarantee $a^2 < b^2$.}{0in}
\question{Suppose $a < b$.
Use examples to check whether $\frac{1}{a} < \frac{1}{b}$.
If you find a counterexample, put a box around it and look for an additional condition which would guarantee $\frac{1}{a} < \frac{1}{b}$.}{0in}
\question{Suppose $a \leq b$.
Use examples to check whether $\frac{1}{a} \geq \frac{1}{b}$.
If you find a counterexample, put a box around it and look for an additional condition which would guarantee $\frac{1}{a} \geq \frac{1}{b}$.}{0in}
\show{If $1 < p$, cite a general result from above to conclude that $5 < 5p$. }{0.2in}
\show{Suppose $p$ is an integer with $-5 < 5p < 5$.
Without dividing by 5, check whether or not it is possible that $p=1$, $p>1$, $p=-1$, $p<-1$.
Conclude that $p = 0$.
If you use properties of inequalities, cite the number from above that you are using.
}{0.5in}
\question{Find the smallest integer $n>0$ for which $\frac{1}{n} < 0.12$.}{0in}
\question{Find the smallest integer $n>0$ for which $\frac{1}{n} < 0.037$.}{0in}
\question{Find the smallest integer $n>0$ for which $\frac{1}{n} < 0.00026$.}{0in}
\question{Let $a > 0$. Describe a procedure for finding the smallest integer $n>0$ for which $\frac{1}{n} < a$.}{0in}
\vfill % pad the rest of the page with white space