Solving linear inequalities is very similar to solving linear equations.
That is we follow the rule, whatever we do to one side of the inequality we do to the other.
By the end of this section you should be able to: ● solve linear inequalities with one variable
The aim of solving a linear inequality ( < > ≤ ≥ ) with one variable is the same aim as for equations with equals signs: to isolate the variable on one side of the inequality.
Worked example Solve the following inequalities and show the solution on a number line. (i) 5x − 2 < 18 where x∈ℕ
Solution
On our course we will only solve inequalities where ax + b ≤ k , where a ∈ ℕ and b and k ∈ℤ .
(ii) 4a+ 6 ≥ 2 where a∈ℤ
Practice questions 32.2
1. Solve the following inequalities and show the solution on a number line.
(i) 3x − 8 > 16 where x ∈ℝ (ii) 2x − 1 < 9 where x ∈ ℕ (iii) 3y − 4 > 17 where y ∈ℤ (iv) 3b + 4 ≤ 13 where b ∈ℝ
2. Solve the following inequalities and show the solution on a number line.
(i) x + 1 ≥ 5 where x ∈ℝ (ii) 6y − 5 < 7 where y ∈ ℕ (iii) 4x + 3 ≥ − 1 where x ∈ℤ (iv) y − 2 ≥ 6 where y ∈ℝ
3. Solve the following inequalities and show the solution on a number line.
(i) 6z + 7 ≤ 1 where z ∈ℝ (ii) y − 6 ≤ 1 where y ∈ℤ (iii) 3x + 29 < 83 where x ∈ℤ (iv) 5y + 38 ≥ 3 where y ∈ ℕ
4. Solve the following inequalities and show the solution on a number line.
(i) 4z + 7 ≤ 27 where z ∈ℤ (ii) 5z − 6 ≥ − 1 where z ∈ℝ (iii) 3x − 5 ≥ − 2 where x ∈ ℕ (iv)
− 7 + 2x ≤ 1 where x ∈ ℕ
5. Solve the following inequalities and show the solution on a number line.
(i) x + 3 > − 6 where x ∈ ℕ (ii) 5z + 8 > − 7 where z ∈ ℕ (iii) 3x + 16 ≥ − 20 where x ∈ℤ (iv) 4x − 40 > − 8 where x ∈ℝ
6. Write down the smallest three integer values of x that make this inequality true.