C2.3 Solve inequalities that involve addition and subtraction of whole numbers up to 20, and verify and graph the solutions.
Skill: Solving Inequalities, Verifying and Presenting Solutions Using Models and Graphical Representations
To facilitate the learning of the concept of inequality, it is important to provide students with activities that encourage them to analyze situations of inequality and to treat them algebraically. It is then essential to discuss with them the strategies used to analyze inequalities, emphasizing those that call on concrete and semi-concrete representations, and that focus on the meaning of the inequality rather than on the mechanical application of a procedure or tedious calculations.
The strategy of plotting solutions with a number line allows students to analyze an inequality using their sense of number, operations and symbol, and to find the range of valid values in an inequality situation.
Students need to consolidate these strategies, as they are the basis for a good understanding of the algebraic manipulations they will be taught in later grades, and can also use these strategies to solve simple equations.
In the example below, each of these strategies is first presented in the context of developing a sense of inequality.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 200.
Represent Solutions Using a Number Line
This strategy involves carefully reading the given inequality and replacing the variable to find the range of valid values. A table is used to find several values for the variable. Afterwards, the solution can be plotted on a number line.
Example
18 - n > 9
The first column in the table represents the number by which the variable n will be replaced in the algebraic expression; for example, 18 - n.
The second column in the table represents the solution of the algebraic expression when the variable n is replaced by the number in the first column; for example, 18 - n, 18 - 5.
The third column in the table confirms or refutes the validity of the value of the variable n; for example, is 18 - 5 > 9?
| 18 | n | > 9 |
|---|---|---|
| 18 | 0 | yes |
| 18 | 1 | yes |
| 18 | 2 | yes |
| 18 | 3 | yes |
| 18 | 4 | yes |
| 18 | 5 | yes |
| 18 | 6 | yes |
| 18 | 7 | yes |
| 18 | 8 | yes |
| 18 | 9 | no |
| 18 | 10 | no |
The range of valid values can be represented using a number line.
The solution is therefore n = {0, 1, 2, 3, 4, 5, 6, 7, 8}.
A number line shows the range of values that hold true for an inequality. An open dot on a number line is used when an inequality involves “less than” or “greater than”, and a closed dot is used when it also includes “equal to.”
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Knowledge: Inequality
Inequality: a non-equal comparison of two quantities.
An inequality is represented by various signs including:
< (is less than);
> (is greater than);
≠ (is not equal to);
≤ (is less than or equal to);
≥ (is greater than or equal to).
Examples
|
78 - 43 < 93 + 25 b > 3 × 52 2 + a < 6 |
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.