C2.4 Solve inequalities that involve two operations and whole numbers up to 100 and verify and graph the solutions.
Skill: Solve Inequalities and Verify and Present 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.
Representing solutions with a number line allows students to analyze an inequality using their number, operations and symbol sense, and to find the range of valid values in an inequality situation.
Students need to consolidate these strategies, as it is the basis for a good understanding of the algebraic manipulations they will be exposed to in later grades. Students can also use these strategies to solve simple equations.
In what follows, each of these strategies is first presented in the context of developing a sense of inequality.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 200.
Represent Solutions Using a Number Line
This strategy involves carefully reading the given number sentence and replacing the variable to find the range of valid values in the inequality situation. A table of values is used to find several values for the variable. Then, the solution can be represented graphically on a number line.
Example
7y + 31 ≥ 78
The first column in the table represents the number by which the variable y will be replaced in the algebraic expression 7y + 31.
The second column in the table represents the value of the algebraic expression when the variable y is replaced by the number in the first column.
7 (0) + 31
0 + 31
0 + 31 = 31
The third column in the table confirms or refutes the validity of the value of the variable y.
Is 31 ≥ 78? The answer is no.
| y | 7y + 31 | ≥ 78 |
|---|---|---|
| 0 | 31 | no |
| 1 | 38 | no |
| 2 | 45 | no |
| 3 | 52 | no |
| 4 | 59 | no |
| 5 | 66 | no |
| 6 | 73 | no |
| 7 | 80 | yes |
| 8 | 87 | yes |
| 9 | 94 | yes |
| 10 | 101 | yes |
The range of valid values can be represented using a number line:
The solution is therefore \(y ≥ 7\).
Note: A number line shows the range of values that hold true for an inequality by placing a dot at the greatest or least possible value. An open dot is used when an inequality involves “less than” or “greater than”; if the inequality includes the equal sign (=), then a closed dot is used.
Knowledge: Inequality
Comparison relationships between two expressions or two quantities.
Inequality is represented by various signs:
< (is less than);
> (is greater than);
≤ (is less than or equal to);
≥ (is greater than or equal to).
Non-Equality
Relationship between two expressions or two quantities that do not have the same value.
Non-equality is represented by the sign ≠ (is not equal to, does not equal).
Example
\(\ 3 \times 5 + 4 \neq 3 \times(5 + 4)\)
\( 8a \neq 25 \)
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 70.