C2.4 Solve inequalities that involve multiple terms and whole numbers, and verify and graph the solutions.
Activity 1: Solving Inequalities
Solve the inequalities below and graph their solution on a number line.
- \(5x + 3 ≤ 2x + 18\)
Sample Solution
Before I start solving this inequality, I need to check if I can collect like terms on each side. Then I will use the balance model to solve the inequality.
\(\begin{align} 5x + 3 &≤ 2x + 18 \\ 5x - 2x + 3 &≤ 2x - 2x + 18 \\ 3x + 3 - 3 &≤ 18 - 3 \\ 3x &≤ 15 \\ \frac{3x}{3} &≤ \frac{15}{3} \\ x &≤ 5 \end{align} \)
The solutions are whole numbers less than or equal to 5. I can graph this solution on a number line by using a closed dot.
- \(5x + 25 + x ≥ 14 - 10x - 7\)
Sample Solution
To solve this inequality, I will first gather the like terms on each side. Then I will use the balance model to solve the inequality.
\(\begin{align} 5x - 25 + x &≥ 14 - 10x - 7 \\ 5x + x - 25 &≥ 14 - 7 - 10x \\ 6x - 25 &≥ 7 - 10x \\\ 6x + 10x - 25 &≥ 7 - 10x + 10x \\ 16x - 25 + 25 &≥ 7 + 25 \\ 16x &≥ 32 \\ \frac{16x}{16} &≤ \frac{32}{16} \\ x &≥ 2 \end{align}\)
The solutions are whole numbers greater than 2. I can graph this solution on a number line by using an open dot.
Source: translated from En avant, les maths!, 7e année, CM, Algèbre, p. 3-5.
Activity 2: Inequalities
Solve the inequalities below. Graph their solution on a number line and then check them.
- \(4x ≤ -6x + 13 + 7\)
- \(9y - 3y - 14 < 4\)
- \(5z - 2 ≥ 3z + 16\)
- \(- 5n > -11 + 3n - 5\)