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

Jul 25, 2017

$\left[\frac{3}{5} , \frac{7}{3}\right]$

#### Explanation:

$f \left(x\right) = \left(3 - 5 x\right) \left(3 x - 7\right) \ge 0$
First, solve f(x) = 0. Solve the 2 binomials:
3 - 5x = 0 --> 5x = 3 --> $x = \frac{3}{5}$
3x - 7 = 0 --> 3x = 7 --> $x = \frac{7}{3}$

On the number line, plot the 2 end-points $\left(\frac{3}{5}\right)$ and $\left(\frac{7}{3}\right)$.
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.
Close interval $\left[\frac{3}{5} , \frac{7}{3}\right]$
---------------------- 0 ----- $\frac{3}{5}$ + + + + + + + + $\frac{7}{3}$ -----------------------