Question

How do you solve and write the following in interval notation: `(3 − 5x)(3x − 7) ≥ 0`?

Answer 1

`[3/5, 7/3]`

Explanation:

`f(x) = (3 - 5x)(3x - 7) >= 0`
First, solve f(x) = 0. Solve the 2 binomials:
3 - 5x = 0 --> 5x = 3 --> `x = 3/5`
3x - 7 = 0 --> 3x = 7 --> `x = 7/3`

On the number line, plot the 2 end-points `(3/5)` and `(7/3)`.
Since a = -15 < 0. The parabola graph of f(x) opens downward.
Between the 2 real roots, f(x) > 0, as a part of the graph stays above the x-axis.
Answer:
Close interval `[3/5, 7/3]`

---------------------- 0 ----- `3/5` + + + + + + + + `7/3` -----------------------

Note. The 2 end-points are included in the solution set.