# How do you solve abs((2 - 5x)/4) >= 2/3?

Apr 7, 2015

Solution

Modulus function can be removed by replacing with definition.
1.$| \frac{2 - 5 x}{4} | \ge \frac{2}{3}$ can be solved in the following way.

2.$| \frac{2 - 5 x}{4} | = \frac{\sqrt{{\left(2 - 5 x\right)}^{2}}}{\sqrt{16}}$
3.$= \sqrt{\frac{4 - 10 x + 25 {x}^{2}}{16}} \ge \frac{2}{3}$
4.$\frac{4 - 10 x + 25 {x}^{2}}{16} \ge \frac{4}{9}$
5.$25 {x}^{2} - 10 x + 4 - \frac{64}{9} \ge 0$
6.$25 {x}^{2} - 10 x - \frac{28}{9} \ge 0$
The boundary is when x satisfies the above equations. Hence
7.$x = \frac{10 \pm \sqrt{100 + 4 \cdot 25 \cdot \frac{28}{9}}}{50}$
8.$x = \frac{30 \pm \sqrt{2900}}{150}$
9.$x = 0.559011 , - 0.15901$
For the $\ge$ constraint to be satisfied the points have to be on the right side of the curve and the left side of the curve.
The given plot is for when the equation 6 satisfies the value of zero.