C2.4 Solve inequalities that involve integers, and verify and graph the solutions.
Skill: Solving Inequalities and Verifying and Graphing Solutions
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 pictorial representations, and that focus on the meaning of the inequality rather than on the mechanical application of a procedure or tedious calculations.
When multiplying or dividing by a negative integer, the inequality sign must be reversed for the solution to be true. When solving inequalities involving integers, special attention must be paid to the inequality sign to ensure that the condition remains valid. It is important to understand that an inequality has a set of solutions and not a single solution like equality.
For example, when multiplying or dividing by a negative number, the inequality sign must be changed:
- \(-2x < 6 \rightarrow x > -3\)
- We know that 4 < 5, if we multiply the two sides of the inequality by -10, it makes \(-40 > -50\). It is indeed -50 which is the smallest so we must reverse the direction of the inequality.
The solution of an inequality that has one variable, such as \(-2x + 3x < 10\), can be graphed on a number line. On this line, an open dot indicates "less than" or "greater than"; a closed dot indicates "less than or equal to" or "greater than or equal to".
One possible strategy for solving inequalities is to consider the inequality as an equality and then try numbers greater or less than the number solving the equality to determine the range of numbers that are valid answers. It is important to encourage students to correctly represent their solutions with an open or closed dot.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.
When solving an inequality, it is best to group like terms with a variable on the side where the coefficient is highest (most positive), since this reduces the opportunity for error in multiplying or dividing by a negative.
However, it is important to be able to isolate the variable if its coefficient is negative.
Once a student has solved an inequality, getting into the habit of checking its solution by inserting that value into the original inequality is an excellent practice to develop.
Example
For example, for -3m < 6, one must divide each side by -3. When performing the division by a negative, the < symbol must be reversed, becoming >.
We thus obtain \(m > -2\).
The solution can be checked by replacing m by -3, -2 and -1.
- \(\begin{align} -3(-3) &< 6 \\ 9 &< 6 \end{align}\)This inequality is false.
- \(\begin{align} -3(-2) &< 6 \\ 6 &< 6 \end{align}\)This inequality is false.
- \(\begin{align} -3(-1) &< 6 \\ 3 &< 6 \end{align}\)This inequality is true.
Knowledge: Inequality
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
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
\(\begin{align}5 &\neq 5 + 1 \\ (3 \times 5) + 4 &\neq 3 \times (5 + 4) \\ 8a &\neq 25 \end{align}\)
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 70.