Question 1.9: Find the range of values for which the following inequalitie......
Find the range of values for which the following inequalities are true, assuming that x > 0. State the solution in words and indicate the solution on the number line.
(a) 10 < x − 12 (b) \frac{-75}{x} > 15 (c) 2x − 6 ≤ 12 − 4x
(a) 10 < x − 12 → 10 + 12 < x → 22 < x (or x > 22)
The solution states: 22 is less than x or x is greater than 22. Hence the solution is represented by all points on the number line to the right of, but not including, 22, as shown in Figure 1.3.
(b) \frac{-75}{x} > 15
Multiply both sides of the inequality by x. Since x > 0, the direction of the inequality sign
does not change.
−75 > 15x
−5 > x dividing both sides by 15
The solution states: −5 is greater than x or x is less than −5. But we assumed x > 0. x cannot be less than −5 and greater than 0 at the same time. Hence there is no solution, as shown in Figure 1.4.
(c) 2x − 6 ≤ 12 − 4x
2x + 4x − 6 ≤ 12 add 4x to both sides
6x ≤ 12 + 6 add 6 to both sides
6x ≤ 18
x ≤ 3 dividing both sides by 6
But we assume x > 0, hence the solution is 0 < x ≤ 3.
This is represented by all points on the number line to the right of 0 and up to and including 3, as shown in Figure 1.5.
