Integer Solutions to Inequality
Task
What is the sum of all integer solutions to 1\lt (x-2)^2\lt 25?
IM Commentary
This task provides a reasonably accessible instance of illustrating cluster (A-REI.B: Solve equations and inequalities in one variable) without being limited to only linear equalities (which are specifically addressed in A-REI.3).
The inequality in this task can be solved in a multitude of ways, including through the use of a table, by manipulating the inequalities, or by making a substitution of variables. The first is the most straightforward and leads to a solution relatively quickly, especially if a little reasoning about magnitude is done beforehand. Students might also use graphing technology: if the function f(x) = (x-2)^2 is plotted then the graph could be used to identify the integer values satisfying the inequality. The downside to this method is that some experimentation will need to be done to decide on the appropriate domain (the range is effectively given as we are interested in integer values of x where 1 \lt f(x) \lt 25). Because of the wide variety of approaches possible, the task might be productively used in a context where students come up with their own solution, and then share strategies in groups or as a whole class.
This task was adapted from problem #8 on the 2012 American Mathematics Competition (AMC) 10B Test. On that test, taken by 35,086 students, the multiple choice answers for the problem had the following distribution:
Choice | Answer | Percentage of Answers |
(A) | 10 | 4.72 |
(B)* | 12 | 38.01 |
(C) | 15 | 38.40 |
(D) | 19 | 3.55 |
(E) | 25 | 2.66 |
Omit | -- | 12.65 |
Of the 35,086 students: 17,169, or 49%, were in 10th grade; 9,928 or 28%, were in 9th grade; and the remainder were below than 9th grade.
Solutions
Solution: 1 Solution Table
We aim to make a table of solutions for the inequality. We start with x = 0 and then put in positive values of x:
x | (x-2)^2 |
---|---|
0 | 4 |
1 | 1 |
2 | 0 |
3 | 1 |
4 | 4 |
5 | 9 |
6 | 16 |
7 | 25 |
8 | 36 |
9 | 49 |
The only values that work for our inequality are x = 0, 4, 5, and 6. As x grows, (x-2)^2 will continue to grow so there are no more positive solutions to this inequality. For the negative values, we find
x | (x-2)^2 |
---|---|
-1 | 9 |
-2 | 16 |
-3 | 25 |
-4 | 36 |
-5 | 49 |
We stopped at -5 as the values of (x-2)^2 are repeating the values when we plugged in positive x's and so there will be no more negative solutions. Our solutions are x = -2, -1, 0, 4, 5, 6.
Solution: 2 Inequalities
We begin by taking the square root of the inequality in question: \sqrt{1} \lt \sqrt{(x-2)^2} \lt \sqrt{25}.
Solution: 3 Change of variables
Suppose we set y =x - 2. Then with this new variable our inequality becomes 1 \lt y^2 \lt 25.
Integer Solutions to Inequality
What is the sum of all integer solutions to 1\lt (x-2)^2\lt 25?