C2.4 Solve inequalities that involve two operations and whole numbers up to 100 and verify and graph the solutions.
Skill: Solving Inequalities and 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.
Representing 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
\(7y + 31 ≥ 78\)
- 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 solution 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 with a number line:
The solution is therefore \(y ≥ 7\).
Note: On a number line, an empty point indicates a strict inequality relationship ("is less than" or "is greater than"); a full point indicates a non-strict inequality relationship ("is less than or equal to" or "is greater than or equal to").
Knowledge: Inequality
The relationship of order between two expressions or two quantities.
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).
Non-Equality
The relationship between two expressions or two quantities that do not have the same value.
Non-equality is represented by the sign ≠ (does not equal, does not equal).
Example
\(\begin{align} 5 &≠ 5 +1 \\ (3 \times 5) + 4 &≠ 3 \times (5 + 4) \\ 8a &≠ 25 \end{align}\)
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 70.